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The Characters and Commutators of Finite Groups

Permanent Link: http://ufdc.ufl.edu/UFE0024770/00001

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Title: The Characters and Commutators of Finite Groups
Physical Description: 1 online resource (67 p.)
Language: english
Creator: Bonner, Timothy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: character, commutator, finite, group, taketa
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Let G be a finite group. It is well-known that the elements of the commutator subgroup must be products of commutators, but need not themselves be commutators. A natural question is to determine the minimal integer, lambda(G), such that each element of the commutator subgroup may be represented as a product of lambda(G) commutators. An analysis of a known character identity allows us to improve the existing lower bounds for |G| in terms of lambda(G). The techniques we develop also address the following related question. Suppose we have a conjugacy class, C, of a finite group, G, such that < C > = G = G'. One may ask for the minimal integer, cn(C), such that each element of G may be expressed as a product of cn(C) elements of the conjugacy class. Again, we improve the known upper bounds, this time for cn(C). Our second focus is the relation between the derived length of a finite solvable group and the cardinality of the set of character degrees in the same group. Over the past few decades, this topic has been explored by Isaacs, Gluck, Slattery, and most recently, by Thomas Keller. There is a standing conjecture that universal constants C sub 1 and C sub 2 exist such that for any finite solvable group G, dl(G) < or = C sub 1 log (|cd(G)|) + C sub 2. Indeed, Thomas Keller has reduced the conjecture to the case of p-groups, and proceeded to attack this case by a study of normally monomial p-groups of maximal class. We extend and refine his methods to a broader class of groups, those for which each irreducible character may be induced from a single normal series. We also examine the special properties held by these groups, said to be normally serially monomial.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Timothy Bonner.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024770:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024770/00001

Material Information

Title: The Characters and Commutators of Finite Groups
Physical Description: 1 online resource (67 p.)
Language: english
Creator: Bonner, Timothy
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: character, commutator, finite, group, taketa
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Let G be a finite group. It is well-known that the elements of the commutator subgroup must be products of commutators, but need not themselves be commutators. A natural question is to determine the minimal integer, lambda(G), such that each element of the commutator subgroup may be represented as a product of lambda(G) commutators. An analysis of a known character identity allows us to improve the existing lower bounds for |G| in terms of lambda(G). The techniques we develop also address the following related question. Suppose we have a conjugacy class, C, of a finite group, G, such that < C > = G = G'. One may ask for the minimal integer, cn(C), such that each element of G may be expressed as a product of cn(C) elements of the conjugacy class. Again, we improve the known upper bounds, this time for cn(C). Our second focus is the relation between the derived length of a finite solvable group and the cardinality of the set of character degrees in the same group. Over the past few decades, this topic has been explored by Isaacs, Gluck, Slattery, and most recently, by Thomas Keller. There is a standing conjecture that universal constants C sub 1 and C sub 2 exist such that for any finite solvable group G, dl(G) < or = C sub 1 log (|cd(G)|) + C sub 2. Indeed, Thomas Keller has reduced the conjecture to the case of p-groups, and proceeded to attack this case by a study of normally monomial p-groups of maximal class. We extend and refine his methods to a broader class of groups, those for which each irreducible character may be induced from a single normal series. We also examine the special properties held by these groups, said to be normally serially monomial.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Timothy Bonner.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Turull, Alexandre.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024770:00001


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THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS


By

TIM W. BONNER




















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009




































2009 Tim W. Bonner









ACKNOWLEDGMENTS

I am sincerely grateful to my adviser, Alexandre Turull. With unyielding patience

and constant support, he has been integral in my academic and personal growth. I

feel indebted to have been his student, and I am fortunate to know such an inspiring

individual.

I also will forever appreciate the encouragement of my family. Over the past six years,

my parents and sister have responded to each step forward and every setback with calm

assurance. Finally, my wife, Emily, has witnessed it all only a glance away, and this work

undeniably bears the steady strength of her hand.










TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... .............. 3

LIST OF TABLES ................... .................. 5

ABSTRACT ................... .................... 6

CHAPTER

1 INTRODUCTION ................... ............... 8

1.1 Products of Commutators ................... ........ 8
1.2 Further Generation Problems .................. ..... 9
1.2.1 Products of Conjugacy Classes . . . . . 9
1.2.2 Products of ('C! i.'ters .. . . . . .... 10
1.3 The Taketa Problem ... . . .......... 11

2 MATHEMATICAL PRELIMINARIES FOR CHAPTERS 3 AND 4 ... .. 13

3 PRODUCTS OF COMMUTATORS AND BARDAKOV'S CONJECTURE ... 17

3.1 A C(' ii i:ter Identity of Burnside . . . . . 18
3.2 Analysis and Results .. . . . ... . .... 21
3.3 Bardakov's Conjecture . . . .. ......... 26

4 PRODUCTS OF CONJUGACY CLASSES AND CHARACTERS .. ... ... 28

4.1 Conjugacy Class Covering Numbers . . . . .. 28
4.2 Comparison with Previous Bounds . . . . .. 33
4.3 C'i o :ter Covering Numbers .. . . . . .... 36
4.4 Comparison with Previous Bounds . . . . .. .. 39

5 MATHEMATICAL PRELIMINARIES FOR CHAPTER 6 .. . . 43

6 THE TAKETA PROBLEM . . . . . . . 48

6.1 Derived Length vs. Number of C('h! :ter Degrees in Certain p-groups 48
6.2 Introduction . . . . . . . . 48
6.3 Normally Serially Monomial p-groups . . . . .... 49

REFERENCES ...... . . . ............. 64

BIOGRAPHICAL SKETCH ... . . .. ............ 67










LIST OF TABLES


Table

4-1 Bound

4-2 Bound

4-3 Bound

4-4 Bound

4-5 Bound

4-6 Bound

4-7 Bound

4-8 Bound

4-9 Bound


Comparison

Comparison

Comparison

Comparison

Comparison

Comparison

Comparison

Comparison

Comparison


for cn(C)

for cn(C)

for cn(C)

for cn(C)

for cn(C)

for ccn(y)

for ccn(X)

for ccn(X)

for ccn(X)


for A 5 . . . . . .

for SL2(5) . . . . . .

for SL2(8) . . . . . .

for Perfect Group of Order 1080 . . .

for M . . . . . .

for A5 . . .

for SL 2 (8) . . . . . .

for P SU3(3) . . . . . .

for M . . . . . .


page

34

35

35

35

36

40

41

41

41









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE CHARACTERS AND COMMUTATORS OF FINITE GROUPS

By

Tim W. Bonner

August 2009

C' I n': Alexandre Turull
Major: Mathematics

Let G be a finite group. It is well-known that the elements of the commutator

subgroup must be products of commutators, but need not themselves be commutators. A

natural question is to determine the minimal integer, A(G), such that each element of the

commutator subgroup may be represented as a product of A(G) commutators. An analysis

of a known character identity allows us to improve the existing lower bounds for |G| in

terms of A(G). The techniques we develop also address the related following question.

Suppose we have a conjugacy class C of a finite group G such that (C) = G = G'. One

may ask for the minimal integer cn(C) such that each element of G may be expressed as

a product of cn(C) elements of the conjugacy class. Again, we improve the known upper

bounds, this time for cn(C).

Our second focus is the relation between the derived length of a finite solvable group

and the cardinality of the set of character degrees in the same group. Over the past few

decades, this topic has been explored by Isaacs, Gluck, Slattery, and most recently, by

Thomas Keller. There is a standing conjecture that universal constants C1 and C2 exist

such that for any finite solvable group G,


dl(G) < C log Icd(G)| + C2.


Indeed, Thomas Keller has reduced the conjecture to the case of p-groups, and proceeded

to attack this case by a study of normally monomial p-groups of maximal class. We extend









and refine his methods to a broader class of groups, those for which each irreducible

character may be induced from a single normal series. We also examine the special

properties held by these groups, said to be normally serially monomial.









CHAPTER 1
INTRODUCTION

1.1 Products of Commutators

The commutator structure of a finite group, G, has been an object of consistent study

since the end of the 19th century. It has been noted by Frobenius [13], that Dedekind

was the first to introduce the idea of a commutator, an element of the group of the form

a-'b-lab, for a, b c G. Dedekind also initiated the study of the subgroup generated by the

set of commutators, later to be called the commutator subgroup. It was soon recognized

that each element of the commutator subgroup need not be a commutator, and such

an element is said to be a noncommutator. William Bejni -ililii Fite [12] was the first to

publish an example of a group containing such an element, though he attributed the

example to G.A. Miller. William Burnside [5], in 1903, subsequently developed a criterion

to determine whether an element of the commutator subgroup was indeed a commutator.

He showed that g c G was a commutator if and only if



X () 0, (1-1)
XEIrr(G)
where Irr(G) denotes the set of irreducible complex characters of G.

This identity can be a powerful tool in determining the existence of noncommutators

and we shall make significant use of a generalization of (1-1). The question then arises as

to identifying of the minimal integer, A(G), such that every element of the commutator

subgroup may be written as a product of A(G) commutators. This invariant A(G) has

been a source of consistent investigation throughout the 20th and 21st centuries. In the

1960's Patrick Gallagher [15] determined an inequality between the size of A(G) and the

order of a finite group, |G|. Later, in 1982, Robert Guralnick [18] demonstrated that for

any positive integer n, one may construct a finite group G such that A(G) = n. He also

determined the minimal finite groups G, with respect to order, such that A(G) / 1 [19].

The famous conjecture of Oystein Ore [38], stating that A(G) = 1 for any finite simple









group G, has been the object of active research through 2008. Contributors to the Ore

conjecture include Gow [17], Ellers and Gordeev [11], and Liebeck, O'Brien, Shalev, and

Tiep [32]. Our work regarding A(G) returns to the earlier considerations of Gallagher. We

improve the known lower bound for |G| for a given value of A(G) and obtain the following

as our first main result.

Theorem A. For I,; finite nonabelian i'. 'air G, we have



|G| > (A(G) + 1)! (A(G) 1)!.

This improves a similar inequality of Gallagher [15] by a factor of 2. Moreover, we use this

result to confirm and strengthen a conjecture of V.G. Bardakov (3.0.18) posed in the most

recent edition of the Kourovka Notebook [36]. Precisely, we obtain the following theorem.

Theorem 1.1.1. Let G be a finite nonabeliam p.i'u;pi. Then, provided CG1 > 1000, we have,


A(G) 1
|G| 250"
We remark that Kappe and Morse [23] have shown that A(G) c {1, 2} for all groups,

G, such that |G| < 1000. Our results regarding products of commutators have been

published in a 2008 volume of the Journal of Algebra [4].

1.2 Further Generation Problems

1.2.1 Products of Conjugacy Classes

Let C be a conjugacy class of a finite group G. Defining C' = {g,... gI gi E C),

one may ask the following questions. Does there exist a value of n such that C' = G,

and if so, what upper bounds can be placed on the smallest such integer? A very famous

related conjecture is that attributed to John Thompson, who r-i---- I 1 that for any finite

simple group G, one may .li- xi- find a conjugacy class, C such that C2 = G. Indeed, one

may show that this implies the Ore conjecture stated in the first section. Returning to our

discussion, it was determined by Arad, Stavi, and Herzog [1] in 1985 that the existence

of an integer n such that C' = G was equivalent to the conjugacy class generating G,









i.e. (C) = G, and G being perfect. In the case that G is simple, very strong bounds have

been obtained, see for example the work of Liebeck and Shalev [33]. We will consider the

general case of G perfect, with the minimal integer called the conjugacy class covering

number and denoted cn(C). In their work, Arad, Stavi, and Herzog [1] obtained av Ii. Iv

of upper bounds on the conjugacy class covering number. Towards the latter end of the

1990's and into the first part of the current decade, David Chillag ([9], [10]) produced

alternate upper bounds for cn(C). In many cases, we improve the known upper bounds of

the conjugacy class covering number, our main result being the theorem below.

Theorem B. Let G be a finite perfect p' -j'r and C a I .-u i:,,1. ;i class of G such that

(C) = G. We /. IU.,' ei(C) = minn GeZ |Z 1 e C>} and for x e Irr(G), we take

X(C) to be the value of x on an element of the v/a, i.o, ;I class C. Then, with m,
{ (c e )l) X e Irr(G) we have bound,



cn(C) < (me, 1) ei(C).

1.2.2 Products of Characters

We now consider a situation regarding the character theory of a finite group which is,

in some sense, dual to that of the generation of a finite group by a particular conjugacy

class. Given X E Irr(G), one may ask for the minimal integer, denoted ccn(x), and called

the character covering number, such that [xccn(x), 0] / 0, for any 0 e Irr(G). David

Chillag [8] showed the existence of ccn(x) if and only if Z (X) = 1G. Further, as with

cn(C) one may determine upper bounds on the invariant ccn(x). In 1989, Zvi Arad and

Hinnit Lipman-Gutweter [2] obtained bounds similar to those of Arad, Stavi, and Herzog

regarding conjugacy class covering numbers. David Chillag ([9], [10]) also obtained bounds

for ccn(x) which paralleled his work with conjugacy class covering numbers. We again,

in many cases, improve the known upper bounds on ccn(x), by an analysis of a character

identity. Our main theorem is the following.









Theorem C. Let G be a finite pi.' r and X c Irr(G) such that Z(X) = 1c. Given C a

I "..I i,. ;i class of G, we take X(C) to be the value of x on an element of C. With ei(X) =

min {nE Z>o [1G, X"'] / 0} and m, (X1) = X( (C(X) I C is a un. ; class of G C

we have,



ccn(X) < (m, 1) *ei(X).

We will complete our work on cn(C) and ccn(x) by demonstrating concrete examples

where our work improves the known upper bounds.

1.3 The Taketa Problem

The second emphasis of our work also examines the connection between the

irreducible complex characters and the commutator structure of a finite group. Here,

we consider the relation between the derived length of a solvable group and the cardinality

of the set of character degrees of the same group. This problem has a long history and a

number of key contributions. In 1930, Ken Taketa [40] proved that for monomial groups,

the derived length of the group, denoted dl(G), is less than or equal to the size of the

character degree set, denoted |cd(G)|. It was conjectured that a stronger, logarithmic

bound held for all solvable groups. Isaacs [20] first proved dl(G) < 3 |cd(G)| 2 for

any solvable group G in 1975. One year later T.R. Berger [3] proved dl(G) < |cd(G)|

in the case |G| is odd, and then in 1985, David Gluck [16] showed dl(G) < 2 cd(G)|

in the general solvable case. Towards the end of the 1990's, Thomas Keller picked up

the problem and in a series of papers ([24], [25], [26]), he reduced the conjecture to a

consideration of finite p-groups. Slattery [39] has shown dl(P) < Icd(P)| 1 for a

p-group, P, such that cd(P) = {l,p,p2,... ,} and Moret6 [37] improved Slattery's

result, establishing a logarithmic bound in terms of the exponent of the maximal character

degree. In 2004, Keller [27] considered the case of normally monomial p-groups, P, of

maximal class, and in this setting, he proved dl(P) < 4 |cd(P)| + 1.









We investigate here a more general family of p-groups. They are defined by the

existence a normal series, P = P > P > ... > Pn- > Pn, such that an arbitrary

irreducible character may be induced from a linear character of one of the subgroups

in the series. A group in this family is said to be n i .irll; i --1. 1,,/ll monomial, and all

normally monomial p-groups of maximal class are normally serially monomial. Our main

result is the following.

Theorem E. Suppose P is a noi iniiul ,.ill;l monomial p-u.'* r, with normal series of

induction,



P= Po > PI > P. > P >-> P.= Ip.

Then, for i C {1,..., n}, {X e Irr(P) X(1p) pi} [Pi:p]

This generalizes and refines an earlier result of Keller [27] in which he determines

the set of character degrees (but not the multiplicity) of a normally monomial p-group of

maximal class. With an additional hypothesis, one may use the above theorem to obtain a

logarithmic bound.

Theorem F. Suppose P is a n. nimall;J ,.:ll monomial p-group, with normal series of

induction,



P = PO > Pi > ... > P.-1 > P= Itp.

Suppose there exists b CE Z such that for i c {0,..., n} and P[ / Ip, we have P[ > P+b.

Then,



dl(P) < 1 + log2 (b |cd(P)| b + 2).

There are a number of examples of normally monomial p-groups for which the above

hypotheses hold.









CHAPTER 2
MATHEMATICAL PRELIMINARIES FOR CHAPTERS 3 AND 4

We now give some background for the mathematics to appear in the next two

chapters. Throughout we take G to be a finite group.

Definition 2.0.1. Let a, b c G. We define the commutator of a and b to be the element

a-lb-lab. Our notation for the commutator of a and b is [a, b].

Definition 2.0.2. We define the commutator subgroup of G, denoted G', to be the

subgroup of G which is generated by all of the commutators in G. Explicity, G' =

( [a,b] a, bG ).

Definition 2.0.3. Let A, B C G. Then, we define [A, B] = ( [a, b] I a c A, b e B ).

Definition 2.0.4. A complex representation of G is a homomorphism E : G GL, (C).

We define the character x afforded by 0 in the following manner. Let g E G. We take

X : G C to be the function defined by x (g) = Tr (E (g)), where Tr is the trace map.

Recalling that for n x n invertible matrices, A, B e GL, (C), Tr (AB) = Tr (BA),

we may see that x is constant on the conjugacy classes of G and thus, said to be a class

function. We remark that any complex representation of G may be extended by linearity

to a representation of the group algebra CG, and as a consequence, so may the associated

character X. There is a close connection between representations and modules. For any

given representation B, we obtain a G-module M in the following manner. We take M as

the set of n-dimensional column vectors over C, and for v e M we determine an action of

G on M according to g v = (g)(v). We then ic the representation E is irreducible if the

corresponding module, MA, is irreducible. Also, given an action of G on an n-dimensional

vector space V (turning V into a G-module), we may choose a basis for V, {e1,...,e,n}

and assign for g c G, the matrix (g e I ... g e,). In this manner, we have determined

a representation from the action of G on V.









Remark 2.0.5. Suppose that V is a G-module affording the representation with

associated character 0 and W is a module affording the representation E and associated

character X. If one considers the representation II afforded by the natural action of G on

V E W, the character associated to II decomposes as w = 0 + X. Thus, sums of characters

are characters, and 0 and x are said to be constituents of w. Further, it may be shown

that if 7 is a character, then w may be written uniquely in the form w = ZxeIrr(G) nxX,

where nX CE Z>o

Remark 2.0.6. Suppose that V is a G-module affording the representation with

associated character 0 and W is a module affording the representation E with associated

character X. Take {vi,...,v,.} as a basis for V and {wi,..., w',,} as a basis for W. It can

be shown that the action of G, g (, 0 wj) = (g I 0 g wj), on V 0 W (extending by

linearity), gives rise to a G-module which affords the character is 0 X. Thus, products of

characters are characters.

Remark 2.0.7. Given a c C, we denote the complex conjugate of a by Ja. We state without

proof that for g c G, x(g-1) = x(g). We further remark that for X c Irr(G), the function

y obtained by composing x with the map of complex conjugation is also an irreducible

character.

Definition 2.0.8. Suppose we have a group G of order n. We may consider the group

algebra CG as an n-dimensional vector space over C. We may take as a basis the n

elements identified naturally with the elements of G. Viewing CG as a G-module under

the action of left multiplication by elements of G, we i the representation afforded by

this module is the regular representation and the character, usually denoted p, afforded by

the representation, is the regular character. By considering the action of G on the basis

elements, and recalling the character is the trace of the image of the representation, we

obtain the following formula.












p(g) { g 1
I|G| g =G

We state without proof that p may also be written as p = EXEIrr(G) X(1)X. This can be

used to recall the important equality |G| = Execrr(G) X2 (G)

We now record some relations between the characters that prove helpful in many

situations, including our calculations with commutators. We list them without proof.

Theorem 2.0.9 (Generalized Orthogonality Relations). Let g, h C G. For in;,

Xi, Xj E Irr(G), we have,


Xi V (gh)S j W ) 6 Xi (
gEG (1)
where 6ij is the Kronecker delta function /. I;,u'. by,


1ij =


Applying the above with h = lG, we get obtain the following.

Theorem 2.0.10 (First Orthogonality Relations). Given Xi, Xj Irr(G) and g G,




gEG
Definition 2.0.11. Let a and 3 be class functions on a group G. We define



gEG

Remark 2.0.12. The above map defines an inner product on the space of class functions on

G, and as a consequence of the first orthogonality relation, we may see that the irreducible

characters form an orthonormal set. Thus, given the character 0 and irreducible character

X, X is a constituent of 0 if and only if [0, X] > 0.









Theorem 2.0.13 (Second Orthogonality Relations). Let g, h c G. For ii:, x E

Irr(G), we have,


x (g) x (h)I CG () f if g is conjugate to h
xeIrr(G) 0 otherwise.
Definition 2.0.14. We now define the central characters. Let E be a complex representation

of the group G, and x the corresponding character. If we take z E Z (CG), the image of z

under E is a scalar matrix, and so we may write E(z) = w,(z)I, where ux is a well-defined

algebra homomorphism from Z (CG) to C. Given the conjugacy class K, if we take the

associated conjugacy class sum K, it is clear that K E Z (CG). By the above, we have

E(K) = uw(K)I. Taking the trace on both sides, we obtain, 1(K) = (l)uw(K). As

characters are linear maps and class functions, we have, IKI (4g) = (l)uw(K), where

g c K. The central character is then given on conjugacy class sums according to,


wU(K) -4g)M
X(1)
We conclude by remarking (without proof) that the conjugacy class sums form a basis

for Z (CG), and so the above formula completely determines the central character as a

function on Z (CG).









CHAPTER 3
PRODUCTS OF COMMUTATORS AND BARDAKOV'S CONJECTURE

We now address Bardakov's conjecture and bounds on the invariant A(G).

Definition 3.0.15. Let G be a finite group. We denote A(G) to be the minimal integer

such that each element of the commutator subgroup may be written as a product of A(G)

commutators.

A result of Robert Guralnick [19] states that the minimal order of a group with

respect to the property that A(G) / 1 is 96. There are exactly two isomorphism types

with this property and of this order. One of the two is given by,

Example 3.0.16.


G = H x (y) where

H = (a) x (b) x (i,j) x x Qs
2Z 2Z

(Y) -~ a" = b, b" = ab, i = j, j" = ij

For this group, the size of the derived subgroup is 32 elements. One may determine

by counting that the set of commutators contains only 29 elements. Hence, A(G) / 1.

Guralnick [18] further constructed, for any integer n, a finite group such that A(G) = n.

The example is presented below.

Example 3.0.17. Let p be a prime and n a positive integer. Set




H = (xI,..., X2n = [Xi, [Xj, Xk] = 1,1 < i,j, k < 2n) and

N = ([xi, xj] i + j > 2n + 1) .


It may be shown that A(H/N) = n.

We note that in Guralnick's construction, the size of the group increases with

increasing values of A(G). It is natural to consider the relation of A(G) with |G| and our









work was initially motivated by a specific conjecture of this nature by V.G. Bardakov. In

the most recent edition of the Kourovka Notebook [36], Bardakov -i-.--- -- I1 the following.

Conjecture 3.0.18 (Bardakov's Conjecture). Suppose G is a finite nonabelian group.

Then, MG < and the inequality is attained only at the symmetric group on 3 letters.

We prove a strengthened version of Bardakov's conjecture. Our proof uses character

theoretic considerations, and largely continues a path outlined by William Burnside and

Patrick Gallagher.

3.1 A Character Identity of Burnside

The primary tool in placing bounds on A(G) is the following lemma whose origins lay

in the work of William Burnside.

Lemma 3.1.1. Given 1: ; finite p. 'pr G, there exists g c G' such that g cannot be written

as a product of n commutators if and only if,



S r() 0 0o < n. (31)
X2j-1 tG)
xeIrr(G)
We note the history of the proof of the above lemma. The case n = 1 was proven by

William Burnside [6] in his text 'The Theory of Finite Groups,' and the case n = 2 was

posed as an exercise. In 1962, the general case was formally proven by Patrick Gallagher

[14].

Proof of the identity. We present the proof given by Patrick Gallagher [14] in 1962,

however, we note that Gallagher proved it in the more general sense of compact (possibly

infinite) topological groups with integrals using Haar measure as opposed to finite sums.

We begin with two lemmas.

Lemma 3.1.2. Let X E Irr(G) and a,gr E G. Then, we have,









Proof. Let E be the representation affording X. For an element g C G, we will define Kg to

be the conjugacy class such that g c Kg, and TK the corresponding conjugacy class sum.

Further, for X c Irr(G) we take uw to be the corresponding central character. We have,


1
X (' [ ])
6E' G


1
t TC r ( E E ( 7 'T T) )
6EG


I Tr -

r (B (U- 1) KLX(T))
1
T wx(T)Tr (B (UT-1))

1 lK lx(T) -1
|K,| X(1G)
X(1G)
S( X(T) 7(T X ((7T 1). )

X(1G)



We now use this result as the base step in an inductive proof of the following lemma.

Lemma 3.1.3. Let X c Irr(G) and a c G. Then,



2n y X(7 [71, 01] ... [rn, -

Proof. First, we prove the case n = 1. We have,




G X12[Ti i])= j 5G X (U7[i,) i])
l i,TiEG T1 EG 1 EG

G1 U(1 X(t)
'M^ ^^ERG









We note the last equality is provided by the generalized orthogonality relation. Now, we

have,




2G > .( .7[T[..]...[T. 6.l 1)




IG 12n 2 p Xr (1)2 X (7 71 mr [1 oi)
61 .l,... -l T1 E,...,T7-l EG
X(u)
2n(1)

The last step follows by induction.

We complete the proof of (3-1) by proving a more general theorem. We count the

number of v-iv an element g in G may be represented as a product of n commutators. We

note g may have multiple representations as a product of n commutators. For example,

given a positive integer n, the identity has multiple representations as a product of n

commutators.

Theorem 3.1.4. Let G be a finite nonabelian i' "'r and g c G. Then, the number of

ways g ii',;l be written as a product of n commutators, denoted by M(g), is given by the

following formula.



M(g) |G|2-1 X (g)
XEIrr(G)
Noting the identity may be expressed as an arbitrary number of commutators, the

above result implies the identities of Gallagher and Burnside in (3-1).

Proof. We recall that the regular character p may be expressed as p = YXEIrr(G) X(lc)X

and recall that the values of p on G are given by,












pG1 9{ 1G
[|G| -g =1


We have,


M(g) = P [Ti ... .])
tIcYg O il ...,[ ,...,1)eG




1 Irr(G) T ,..., ..., G

SY (() 2n by Lemma (3.1.3),
xEI Irr(G)

= |G|2n-1
=1 x| 2n-1
xE Irr(G)
xEIrr(G)


|1G|n-(1 2 1
XEIrr(G)


So, the theorem is proved.

3.2 Analysis and Results

The following theorem is our primary tool in establishing a relationship between A(G)

and |G|. It is implicit in the work of Gallagher [15] and we state it here in a way more

convenient for our purposes.

Theorem 3.2.1. Let G be a finite nonabelian i"-q' with character degrees 1 = fo < fi <

.. < fr. Set Ai = '. Then A(G) < r. Further, if 1 < n < r, and p(x) c C[x] of degree n,

p(l) / 0, and,



I p(A1) p(1A,)> C


then A(G) < n.









Proof. Suppose the theorem is false. For each polynomial p(x) such that p(l) / 0, set

mp = min P ..., There exists an n and polynomial p(x) of degree n such

that p(l) / 0 and either n = r, or n < r and mp > G'|. Among all such n, pick n as small

as possible. Note, n > 1 and A(G) > n. It follows that there exists g C G' such that g

cannot be written as a product of n commutators. By Lemma 2.1, we have



t -1 1x() = 0 (0 < k < n). (3-2)
xeIrr(G) X(1)
Setting


| G |
ai= -X X (g)(
xeIrr(G), x(l)-= i
we rewrite the system of equations in (3-2) as,



i0 (0 < k < n),
i=0
with ao = 1. It follows that if q(x) c C[x] of degree at most n, we have,



Y aiq(A) 0. (3-3)
i=0

Take q(x) = H (x Aj). Now, Ao = 1, and Ai < 1 for i > 1, imply that q(Ao) / 0 while
j=1
q(Ai) = 0 for 1 < i < r. This, together with equation (3-3), implies that n < r and
mi > CG'|. Set .1,. = max{|p(Ai)| ,..., |p(Ar) }. Note .1,. > 0, as r > n. By (3-3), we

know that,

ap(Ai) -= p(l)|. (3-4)
i= 1
We have,












xEIrr(G), x(l)>l i-1

> G a|p(Ai)
S-l1 Il il^)
SIG ) by (3-4)

|G|


On the other hand,



S xW~x~g)\5x(1= > l() 5) x()xg)l
xeIrr(G), x(l)>l xeIrr(G) xeIrr(G), x(1)=-

< 5 ^(1)- 2 x(l)x(g)
xeIrr(G) xeIrr(G), x(1)=-


IGI IGI
|G (G' 1) < |G1 mP,

since we are under the hypothesis mp > |G' This is a final contradiction and the theorem
holds.

Now, we determine a specific polynomial to obtain our bounds.
Lemma 3.2.2. Suppose we are given a sequence of integers fo,... fr where 1 = fo <

fi < f2 < ... < fr, and some n such that 1 < n < r. We set Ai = 7Z. Then, there exists
p(x) cE R[x] of degree n such that, p(l) / 0 and,


(l) > fi(f- 1) 1 for j 1,2...,r. (3-5)
P-ini) +t q

Proof. We set q(x) = H (x Ai) and c (l)n We take p(x) = c + (1 c) .
i=1 2 n (f2-1)+(-1)n+1
For 1 < j < n, p(Aj) = c, and p(l) = 1, so the inequality in (3-5) holds for 1 < j < n.









As p(x) is of degree n and constant on the set {Ai,..., An}, p(x) must be monotonic on


(-oo, A,). For n < j < r, O < A, < An, and we have either,


or p(A,) > p(Aj) > p(O).


This gives,


|p(Aj)|

p(l)


() p() p(l)
p(0) p(A,)


We adopt the convention that (3-6) holds if p(Aj)


q(1)
q(0)


i)(- nf (
i 1


for n


for n < j< r.

0. We have,


( C)


This gives p(O)


c+ (1 c) i
l ^ q(1)


c. As p(l) = 1 and p(A,)


c, (3-6) implies that the


lemma holds.



Corollary 3.2.3. Let G be a finite nonabelian ii ;"-' with character degrees 1

... < f, and suppose 1 < n < r. If


n
2 l(f -1)


fo < fi <


1> G' ,


then A(G) < n.

Proof. Set A, -

p(l) / 0 and,


7. By Lemma (3.2.2) we may find p(x) c R[x] of degree n, such that
it


p( > 2 (f -1)
P(j niI 1


for j 1,2...,r.


Then, Theorem (3.2.1) gives the result.

Corollary 3.2.4. Let G be a finite nonabelian ii ;"-' with character degrees 1


... < f,, and suppose 1 < n < r. If


(3-6)


fo < fi <


p(A,) < p(Aj) < p(0)













i= 1


then A(G) < n.

Proof. Now,


i (f2 n)


Hi1


n+1
) > f(i
'i i-2


Sn
(1 + )
i= 1


n 2 n
2 (2 1) > ,
i= 1 i= 1
as f, > n + 1. Then,


2fl(f -1)
i= 1


n
i > f + 1
i=1


n

i-1


and the result follows from Corollary (3.2.3).

For any finite nonabelian group G, Gallagher [15] showed that


IG' > 1/2 (A(G) + 1)! (A(G)- 1)! + 1.


In the case G is a p-group, with IG'|

strengthen each of these inequalities.


p", he also showed that a > A(G) (A(G)


Corollary 3.2.5. Let G be a finite nonabelian /i -ior' and suppose n


A(G) > 1. Then


IG' > (n + )!(n -1)!.

Proof. As A(G) > n 1, Theorem (3.2.1) shows the G has at least n + 1 distinct character

degrees. Further, Corollary (3.2.3) implies that,


n-1
G'| > 2 (f
i= 1
and the result follows.


n
t) t > 2 (2 _t


1= (n + 1)! (n


n+2
n+t


(3-7)


1). We


1)! 1,









We note that the group of minimal order such that A(G) / 1 has 96 elements. For this

group, Gallagher's inequality shows that A(G) E {2, 3} while our inequality shows that

A(G) = 2.

Corollary 3.2.6. Let G a finite nonabelian p-_i,' r", |G'| = pa, and A(G) = n > 1. Then

a > n(n 1).

Proof. As G is a p-group, each character degree is a power of p. As A(G) > n- 1, Theorem

(3.2.1) shows that G has at least n + 1 distinct character degrees. Then, Corollary (3.2.4)

implies that,
n-l n-l
pa G'| > f? > p2i (n-1)
i 1 i 1
and the result follows.

Corollary 3.2.7. Let G be a finite nonabelian ii with character degrees 1 < f, < f2.

Then A(G) < 2, and if A(G) 2, IG' > L2ff 2 f f
f12 f1

Proof. Set A =- for i E {1,2}. By Theorem (3.2.1), A(G) < 2. Let p(x) be the line

through ( 0) and (1, 1). We have I= i(= 2)f f, so Theorem (3.2.1)

gives the result.

Remark 3.2.8. For a group with exactly 3 character degrees, this is the best possible

bound using the method of Theorem (3.2.1). We note there exists a group G such that

|G| = 128, the set of character degrees is {1, 2, 4}, and G' contains a noncommutator. For

this group, Corollary (3.2.6) gives IG'I > 8, Corollary (3.2.7) gives IG'I > 10, and the

actual value of I G' is 16.

3.3 Bardakov's Conjecture

We now answer the question of Bardakov.

Corollary 3.3.1. Let G be a finite nonabelian iq'. GThen M) < 1 with the bound

obtained only at the 'i .''i S3. FurtherT, limeco ), = 0, and for i;ii 'i.''.q such that

G > 1000, we have, A(G) < .
|G| 250









Proof. By Corollary (3.2.5), we have,

A(G) A(G) < A(G) 1
C|G| |G' (A(G) + 1)!(A(G) 1)! (A(G) + 1) (A(G) 1)!2'

So, < i if A(G) > 3. As A(G) = 1 for all groups G where |G| < 12, the first statement
follows. We let c.d.(G) represent the set of character degrees of G. Then, Theorem (3.2.1)

gives,
A(G) c.d.(G)| < 1
IG| I|G |G| G
and we have the limit in the second statement. Finally, suppose |G| > 1000 and (G) >
20. Then A(G) > 4. By (3-8), < (G) and the last statement results.

Corollary 3.3.2. The -i o /, i of Bardakov is true.

Proof. This is clear by Theorem (3.3.1).

Remark 3.3.3. Recently, L-C. Kappe and R. Morse [23] implemented the system of

equations in (3-1) in GAP to show A(G) < 2 for all groups G such that |G| < 1000.









CHAPTER 4
PRODUCTS OF CONJUGACY CLASSES AND CHARACTERS

Let C be a conjugacy class of a finite perfect group G such that (C) = G. In their

1985 work, Z. Arad, J. Stavi and M. Herzog [1] demonstrate the existence of a positive

integer n such that C" = G. The minimal positive such integer is called the conjugacy

class covering number, and denoted cn(C). The conjugacy class covering numbers of

numerous groups, including the alternating groups, finite groups of Lie type, and special

linear groups have received study ([33], [28]). It is natural to ask, given any conjugacy

class C of a finite perfect group G where (C) = G, what upper bounds may be placed

on cn(C)? Work has been done on this query, with the first bounds obtained in the

noted work of Arad, Stavi, and Herzog. Thereafter, in an approach using semisimple

commutative algebras, David Chillag [9] shows that if m = J$ c Irr(G) then

cn(C) < m2 2m + 2. In a further analysis, Chillag [10] obtains an alternate bound,
showing that with k the number of conjugacy classes of G, cn(C) < (fl) (m 1). Similar

to our work with products of commutators, we will conduct an analysis of a character

identity and obtain upper bounds which in many cases improve the existing ones.

4.1 Conjugacy Class Covering Numbers

We begin with the relevant definitions.

Definition 4.1.1. Let G be a finite group and C a conjugacy class of G. We set C" =

0{g ...g I gi,..., g, e C}.
Definition 4.1.2. Let G be a finite group and C a conjugacy class of G. We set ei(C) =

min {n Z+ | 1 E CC }.

Remark 4.1.3. Let g c G where g c C. Then, ei(C) < o(g) where o(g) is the order of g.

Definition 4.1.4. Let G be a finite group and C a conjugacy class of G. Given x e Irr(G)

and C a conjugacy class of G, we take X(C) to be the value of x on an element of C. We

set m(C) x= ( | ex Irr(G)} and m(C) {= ( ) e)(c) X C Irr(G)}.
X(IG) ( X(C)









Our work derives principally from an analysis of the following character identity,

which may be found in Arad, Stavi, and Herzog [1].

Theorem 4.1.5. Let G be a finite group, C a .'i in,. ', class of G. Let g c G. Then

g c C" if and only if
X(C)"n(g- 1)
X(1)"-1
XEIrr(G)
Using a technique similar to that in our work on products of commutators, we

establish the following.

Theorem 4.1.6. Let G be a finite perfect nonabelian jii;-'r and C a I .'i i',. ;/ class of G

such that (C) = G. Set mi = me, (C)| and enumerate me,(C) = {co,. ,ame -1 } where

ao = 1. Then, cn(C) < (m, 1) ei(C). Further, if 1 < n < m,, 1, and p(x) E C[x] is a

I 1111.;*in'l,: ,1 of degree n, p(l) / 0 and,


min PM p ') > G1,
p(1) p()am-)

then cn(C) < n el(C).

Proof. Suppose the theorem is false. For each polynomial p(x) such that p(l) / 0, set

m =- min .. p(a ,, 1)-) There exists an n and a polynomial p(x) of degree n

such that p(l) / 0 and either n = m, 1, or n < me, 1 and m, > |G1. Among all such

n, pick n as small as possible. Note, n > 1 and cn(C) > n ei(C). It follows that there

exists g c G such that g i Cnel(c). By the definition of ei(C), we have,

1c C Ce(C) C2c(C) C3e1(C) c ...

We adopt the convention that Co = 1 and X(C)o = 1 if x(C) = 0. Then, noting the values

of the regular character for the case k = 0 and employing Theorem (4.1.5) for 1 < k < n,

we have,

0 0 < k < n. (4-1)
xrr( (1)ke(C)-1
XEIrr(G)










Setting



xEIrr(G), ( y) (c)=-a

the system of equations in (4-1) becomes,

me1 -1
Z aa,- =0 (0 < k < n).
i=0

It follows that if q(x) is any polynomial of degree at most n, we have,


e, 0-1.
Y aiq(ai) = 0.


(4-2)


Let x C Irr(G). If ( )x( c) 1, then IX(C)| -= (1) so C C Z(X). So, as G = (C)

and G is nonabelian, X = 1G, and ao = 1. Take q(x) = ] (x a,). Now, co = 1, and
j=1
asi / ao for i > 1, imply that q(ao) / 0 while q(ai) = 0 for 1 < i < m, 1. This,

together with equation (4-2) and ao = 1, implies that n < m,, 1 and mp > |G1. Set

1,. = max {|p(ai)| ,..., p(cime,-i) }. Note .1,. > 0, as m, 1 > n. By equation (4-2), we

know that,


Tme -1
Saip(a ) -|p(l)|.
i 1


XEIrr(G), x(1G)>l


me -l


*, aip(ai)
i= 1

> P by (4-3)

> mp.


Alternatively,


We have,


(4-3)














XEIrr(G), x(1G)>l


x(iG)x(g-'1)


ye rr(G) X(g1)
XEIrr(G)


XEIrr(G), x(1)-1


x(iG)x(g-'1)


< Y X()2
XEIrr(G)


XEIrr(G), x(1G)=1


-IGI 1

< m,,


since we are under the hypothesis m, > IGI. This is a final contradiction and the theorem

holds.

Lemma 4.1.7. Suppose we are given a sequence of complex numbers ac,... cmer 1 where

Qai+1| < jail < 1 for i = 1,..., mn 2 and some n such that 1 < n < m,, 1. Then, there

exists p(x) E C[x] of degree n such that p(l) / 0 and,


p(l) > 1

H (x ai )I

Proof. C('!h.-. p(x) -- So, p(l) -
n (4-4) holds for <
Then, the inequality in (4-4) holds for 1 <


for j = 1,2,...,me,


1. We adopt the convention that |


j < n. For j > n,


_P() It__1 -_ ail_|

i 1
>n 1 n n

2|ail 2TH |a| 2n
>^^->^ ^--^ii


So, the inequality in (4-4) is true for 1 < j < m,, -

This immediately gives the following corollary.


(4-4)


00.









Corollary 4.1.8. Let G be a finite perfect ii ";'r and C a ;. i i,'. ;I class of G such that

(C) = G. Enumerate m,,(C), {cao, ame-i}, where ao = 1 and I|a+i| < aI i < 1 for

i = ,... me 2. If, for some l < n < m, 1 we have,




i= 1
then cn(C) < n el(C).

Proof. By Lemma (4.1.7), we may find a polynomial p(x) E C [x] of degree n, such that

p(l) / 0 and,



p(j) > ( 1-) for 1,2..., .

Then, p(x) and Theorem (4.1.6) give the result.

We now give an interesting bound for cn(C) in group theoretic terms.

Corollary 4.1.9. Let G be a perfect i -.;"q, and C a I un i i,' ;I class of G such that

(C) = G. Let g be an element of C and o(g) the order of g. Then,


cn(C) < o(g) (|Cc(g)| + 1).




Proof. From Theorem (4.1.6) we have the bound cn(C) < (me 1) ei(C). We denote

n(g) = |{X E Irr(G) | (g) = 0}|. This gives me,,(C)| < Irr(G)| n(g) + 1. It is
a lemma of Gallagher [14], that n(g) > |Irr(G)| |Cc(g)|. This gives us the bound

me'(C)| < |Cc(g) + 1. Now, we note that ei(C) < o(g) and the result follows.

Modifying an argument of S. Garrison (Problem 4.2 in [21]), we determine an upper

bound on ei(C). This will enable us to place an upper bound for cn(C) in character

theoretic terms.









Lemma 4.1.10. Let G be a finite perfect nonabelian i'i r and C a ;./', i',i. class of G

such that (C) = G. Set m = |m(C)I and enumerate m(C) = ao,...a m-1 where ao = 1.

Then, ei(C) < m.

Proof. Let C-1 be the inverse class of C. We claim that C-1 C C for some s with

1 < s < m 1, and the result will follow. Suppose the claim is false. Then, C-1 9 Ck

where 1 < k < m 1. Recalling the values of the regular character for k = 0 and using

Theorem (4.1.5) for 1 < k < m 1, we have,


E () 0 0 < k < (4-5)
XEIrr(G)

Setting

ai= ()X(CcG),
EIrr(G), X(C)
X(OG)
the system of equations in (4-5) becomes,

m-1
-lai = 0 (0 < k < m 1). (4-6)
i=0

Now, the matrix (a- 1) is a Vandermonde matrix, so det (ak-1) / 0, and equation (4-6)

has only the trivial solution. However, we note that if () = 1, then C C ker As
x(IG)
G = (C), we have X = 1G. This implies ao = 1, which gives a nontrivial solution to

equation (4-6), a contradiction. So, C-1 c C" for some s with 1 < s < m 1, and hence,

ei(C) < m.



Lemma (4.1.10) together with Theorem (4.1.6) yield the following bound on cn(C).

Corollary 4.1.11. Let G = G' and C a -.'i i .i,.- ;/ class of G such that (C) = G. Set

m = m(C) and m, = me, (C) 1. Then, cn(C) < (m,T 1) m.

4.2 Comparison with Previous Bounds

In 1985, Arad, Herzog, and Stavi [1] established that given a finite perfect group G, in

which (C) = G, with k the total number of conjugacy classes, A the number of nonidentity









real conjugacy classes, and p equal to { the number of complex conjugacy classes,

cn(C) < (A +) e(C). (4-7)

Then, in 1997, in the same situation with m= |m(C)|, David Chillag [10] showed that,


cn(C) < m2 2m + 2. (4-8)

Subsequently, in 2005, Chillag, Holzman, and Yona [10] proved,


(Irr(G)|t
cn(C)<(m 1)(r 1 +). (4-9)

To give evidence for the improvements that can be made with our bounds we will consider

a few small nonabelian perfect groups where computations can be efficiently performed

with GAP. Specifically, we consider the groups A5, SL2(5), SL2(8), the perfect group of

order 1080, and the Mathieu group, M11. For each group, we will list the conjugacy classes

which generate the group, the actual covering number, and the various bounds discussed

above. We denote the bound in (4-7) as ASH, the bound in (4-8) as Cl, the bound in

(4-9) as C2, our bound of cn(C) < ei(C) (me, 1) as Bl, our bound in Corollary (4.1.11)

as B2, and our group theoretic bound in Corollary (4.1.9) as B3. We note the upper

bounds were computed with a program in GAP that takes as input the group in question,

and returns as output the generating conjugacy classes with the respective bounds. The

actual covering numbers were computed with a separate program.

Table 4-1. Bound Comparison for cn(C) for A5
Class Actual ASH Cl C2 B1 B2 B3
2 2 8 10 12 6 12 10
3 2 8 10 12 6 12 12
51 3 8 17 16 8 20 30
52 3 8 17 16 8 20 30









Table 4-2. Bound Comparison for cn(C) for SL2(5)
Class Actual ASH Cl C2 B1 B2
3 3 16 17 24 8 20
4 2 16 10 18 6 12
51 5 16 50 42 14 56
52 5 16 50 42 14 56
6 3 16 26 30 8 24
101 5 16 65 48 14 63
102 5 16 65 48 14 63


Table 4-3. Bound Comparison for cn(C) for SL2(8)


Class
2
3
71
72
73
91
92
93


Actual
2
3
2
2
2
3
3
3


ASH
16
16
16
16
16
16
16
16


C1 C2 B1 B2
10 18 6 12
17 24 8 20
26 30 10 30
26 30 10 30
26 30 10 30
37 36 12 42
37 36 12 42
37 36 12 42


Table 4-4. Bound Comparison for cn(C) for Perfect Group of Order 1080


Class
2
31
32
4
51
52
61
62
121
122
151
152
153
154


Actual
4
3
3
4
4
4
4
4
4
4
4
4
4
4


ASH
22
22
22
22
22
22
33
33
33
33
33
33
33
33


C1 C2
50 70
26 50
26 50
26 50
50 70
50 70
145 120
145 120
82 90
82 90
145 120
145 120
145 120
145 120


B1 B2
10 40
10 30
10 30
10 30
14 56
14 56
21 91
21 91
15 50
15 50
21 91
21 91
21 91
21 91


B3
50
30
30
56
80
80
150
150
156
156
240
240
240
240


The data shows that for each conjugacy class of each listed group, the bound B1

is the strongest. In every case with the exception of the class of elements of order 4 in









Table 4-5. Bound Comparison for cn(C) for Mil
Class Actual ASH Cl C2 B1 B2 B3
2 3 14 50 42 12 48 98
3 2 14 37 36 12 42 57
4 2 14 26 30 10 30 36
5 2 14 17 24 8 20 40
6 3 14 26 30 8 24 42
81 3 21 37 36 18 42 72
82 3 21 37 36 18 42 72
111 3 21 26 30 15 30 132
112 3 21 26 30 15 30 132



SL2(5), the bound in ASH is the next strongest. The value of the bounds Cl and C2

is that a computation of the invariant ei(C) is not necessary, and similarly B3 gives an

upper bound without the use character theoretic information. We note our bound B1 is

still distant from the actual values, and much improvement seems possible.

4.3 Character Covering Numbers

We also consider a dual covering problem with regard to irreducible characters. Given

an irreducible character X, we would like to determine the minimal positive integer n

(assuming existence) such that each 0 E Irr(G) is a constituent of X". In terms of the
inner product, this is the minimal integer n such that [0, X"] > 0 for all 0 E Irr(G).

This minimal number is denoted ccn (x), and is called the character covering number.

In 1987, David Chillag [9] showed the existence of the integer if and only if Z (x) = 1G.

Arad and Lipman-Gutweter [2] continued by providing upper bounds for ccn(X) similar

to those found for conjugacy class covering numbers in the work of Arad, Stavi, and

Herzog [1]. Arad and Lipman-Gutweter also determined that for any X c Irr(G), where

G = Sz(q), q = 22n+1, ccn(x) < 4. Subsequently, similar to his analysis of conjugacy

class covering numbers, David Chillag [9] showed that for c Irr(G) and m(x) =

Sx |(l) C a conjugacy class of G } one has the bound cen(x) < mT()2 2m(x) + 2.
He later showed [10] that if k is the number of irreducible complex characters of G,

ccn(x) < ([) (m(x) 1). We determine upper bounds for cen(x) parallel to our bounds
2I "\l Iv~U~IIII ~~ VIU V ~I/~ CI~II~ U VIU









for cn(C), and in many cases, our bounds prove stronger than the known ones. We will
provide examples at the end of the section.
Definition 4.3.1. Let G be a finite group and X E Irr(G) where Z(X) = We denote

el(X) = min {n Z+ I [1G, X"] / 0}.
Definition 4.3.2. Let G be a finite group and x e Irr(G) where Z(X) = 1. Given

X C Irr(G) and C a conjugacy class of G, again we set X(C) to be the value of x on
an element of C. We further denote m(X) X= I| C a class of G rand mei(X) =

{( )(x C a class of G .
We now have our main theorem regarding character covering numbers. Its statement
and proof are similar to Theorem (4.1.6).
Theorem 4.3.3. Let G be a finite p"i;-'r and X E Irr(G) such that Z(x) = 1. Set

me, = nme,(x) and enumerate me, (X) = ao, .... m -1} where o = 1. Then,

ccn(X) < (m,n 1) e i(). Further, if 1 < n < m, 1, and p(x) is a p* 'i;';,. ';;i.il of degree

n, p(l) / 0 and,


min p() p(1) > GJ

then ccn(X) < n ei(X).

Proof. Suppose the theorem is false. For each polynomial p(x) such that p(l) / 0, set
mp = min p(a p( 1)-, .) There exists an n and a polynomial p(x) of degree n

such that p(l) / 0 and either n = m, 1, or n < me, 1 and mp > |G1. Among all such
n, pick n as small as possible. Note, n > 1 and ccn(X) > n ei(C). It follows that there
exists 0 e Irr(G) such that [10, Xei(x)] = 0. By the definition of ei(X), we have,

1 c Xe(x) X2e1(x) C X3e(x) C ...

We adopt the convention that xo = 1G. We have,


Y Xkel(x)(g) g-) = 0
gEG


0 < k < n.









This yields the system of equations,


( ))kr(X) ) =0 0 \XG(1tG) ) 0G 0 (1) ~ n. (~
yEG

Setting
S09(g-1)



the system of equations in (4-10) becomes,


aa a,- 0 (0 < k < n).
i=0

So, if q(x) is any polynomial of degree at most n, we have,

rnme -1
Y aq(a) 0. (4-11)
i=-O

As Z(X) 1, ao 0 / 0. Take q(x) = fo (x aj). With ao = 1, and ai / ao for
j=-1
i > 1, we have q(cao) / 0 while q(ai) = 0 for 1 < i < m,1 1. This, together with equation

(4-11), implies that n < m,, 1 and mrn > |G1. Set .1,. = max {|p(ai)| ,..., p(ame-1)

Note .l,. > 0, as me, 1 > n. By equation (4-11), we know that,

me1-1
Saip(ai) = p(l) (4-12)
i= 1

We have,



O(1 ) -1 me1-1
0(91) 5n,1 a I r aip(ai)
gEG,gS1c 1 1

> r4PM by (4-12)

> However.


However,












0( 91) 0 V e(G) I t < mp.
gEG, 9S41G gEG, S74IG 1

This is a contradiction and the theorem holds.

We now combine Lemma (4.1.7) with Theorem (4.3.3) to obtain the following

corollary.

Corollary 4.3.4. Let G be a finite ii and X E Irr(G) such that Z(X) = 1. Enumerate

me(X) = {ao,...,a;me -1} where ao = 1 and |ai 1 < Jai, for i = ,...,m 2. If,



i= 1
then ccn(X) < n ei(X).

As before, we would like a bound for el(X). This time, we use the theorem of Brauer

and Burnside regarding powers of faithful characters.

Lemma 4.3.5. Let G be a finite pi ri, and X c Irr(G) such that Z(X) = 1c- Set

mn= nm(x). Then, el(X) < m.

Proof. By the theorem of Brauer and Burnside (Theorem 4.3 in [21]), we know that there

exists 1 < s < m 1 such that [X, x"] > 0. So, [iG, X'1] > 0 and the result follows.

Lemma (4.3.5) in conjunction with Theorem (4.3.3) immediately imply our final result

on character covering numbers.

Corollary 4.3.6. Suppose G is a finite i"i;jr, and X E Irr(G) where Z(X) = 1G. Then,

with m= |m() I and m, = me, (X)|, we have ccn(X) < (m, 1) m.

4.4 Comparison with Previous Bounds

We now compare our bounds with the existing efforts. Given a finite group G,

we denote the number of irreducible characters by k, the number of real, irreducible,

and nonprincipal characters by A, and we denote the number of nonreal irreducible

characters by p. Similar to the bound in the context of conjugacy class covering numbers,









in 1989, Arad and Lipman-Gutweter [2] established that given a finite group G, X c Irr(G)

such that Z(x) = 1G,

ccn(X) < e (X) (A + p). (4-13)

Then, in 1997, in the same situation with m = |m(x)|, David Chillag [10] showed that,

similar to the conjugacy class case,



ccn(x) < m2 2m + 2. (4-14)

Also paralleling the conjugacy class case, in 2005, Chillag, Holzman, and Yona [10]

proved,



ccn() < (m 1) (rr) + ) (4-15)

We contrast our bounds in Theorem (4.3.3) and Corollary (4.3.6) with those above by

examining the character covering numbers of the characters from the groups A5, SL2 (8),

PSU3(3) and the Mathieu group M11. We label the bound in (4-13) ALG, the bound in

(4-14) Cl, the bound in (4-15) C2, and our bounds in Theorem (4.3.3) and Corollary

(4.3.6), B1 and B2 respectively. Again, the calculations were done with a computer

program that accepted the group and returned the values of the bounds. A separate

program computed the actual values of the character covering numbers.

Table 4-6. Bound Comparison for ccn(x) for A5
Degree Actual ALG Cl C2 B1 B2
3 3 8 17 16 8 20
3 3 8 17 16 8 20
4 2 8 10 12 4 8
5 2 8 10 12 4 8









Table 4-7. Bound Comparison for ccn(X) for SL2(8)
Degree Actual ASH Cl C2 B1 B2
7 4 16 17 24 6 15
7 3 16 37 36 10 35
7 3 16 37 36 10 35
7 3 16 37 36 10 35
8 2 16 10 18 4 8
9 2 16 26 30 10 30
9 2 16 26 30 10 30
9 2 16 26 30 10 30


Table 4-8. Bound Comparison for ccn(X) for PSU3(3)
Degree Actual ASH Cl C2 B1 B2
6 6 18 37 48 8 28
7 5 18 37 48 8 28
7 5 27 101 80 30 110
7 5 27 101 80 30 110
14 4 18 37 48 8 28
21 3 18 26 40 8 24
21 3 27 65 64 24 72
21 3 27 65 64 24 72
27 3 18 17 32 6 15
28 3 27 65 64 24 72
28 3 27 65 64 24 72
32 3 27 26 40 15 30
32 3 27 26 40 15 30



Table 4-9. Bound Comparison for ccn(X) for MI,
Degree Actual ASH Cl C2 B1 B2
10 3 14 17 24 6 15
10 4 28 37 36 16 28
10 4 28 37 36 16 28
11 3 14 26 30 8 24
16 3 21 26 30 15 30
16 3 21 26 30 15 30
44 2 14 17 24 6 15
45 2 14 17 24 6 15
55 2 14 10 18 4 8









The bound comparison involving character covering numbers is similar to that of

conjugacy class covering numbers, yet there are some small discrepancies. Our bound, B1,

is usually, but not ahb-i-; the strongest bound. There are two irreducible characters of

degree 7 in PSU3(3) where ASH = 27 and B1 = 30. So, a proof that the bound B1 is

stronger than ASH is not possible, and gives evidence that the same can hold with cn(C).

As before, the examples with character covering numbers show the bounds are most often

far from the actual values. However, there are some instances where the bound B1 stays

close to the actual value of ccn(y). For one irreducible character x of degree 7 for SL2(8),

ccn(x) = 4 while B1 gives 6. For the irreducible character x of degree 6 for PSU3(3),

ccn(x) = 6 while B1 gives 8. Whether this holds with increasing group order is unclear.









CHAPTER 5
MATHEMATICAL PRELIMINARIES FOR CHAPTER 6

We now give background for the chapter concerning the Taketa problem and

normally serially monomial p-groups. We take G to be a finite group and begin with

some customary definitions.

Definition 5.0.1. We define G) = G' and recursively define G") = (G("-1))'. We then

form a characteristic series,



G > G ) > G(2) > G(3)>

This series is called the derived series. If there exists an integer no such that G() = 1G,

we ic that the group G is solvable, and define the derived length of G, denoted dl(G), to

be the minimal such integer no such that G( G

Definition 5.0.2. We set G1 = G and recursively define G, = [G,_1, G]. We form a

second characteristic series,


G > G = G' > G3 > G4 > G...,


called the lower central series of G. If there exists an integer mo such that Gmo = 1G, we

z.- that the group G is nilpotent, and define the nilpotence class of G to be the minimal

such integer mo such that Gmo = 1G-

Definition 5.0.3. We note that a well-studied family of p-groups are those groups P such

that |P| = p"' and whose nilpotence class is n 1. These groups are said to be of maximal

class.

A record of many of the important properties of groups of maximal class may be

found in the group theory text by Leedham-Green and McKay [31]. We shall recall a few

of their properties that will be of subsequent use. If we form the lower central series,



P > P2 > > Pn-1 > Pn = P,










we have that the subgroups Pi for 2 < i < n 1 are the unique normal subgroups of P of

index p'. We define Pi = Cp (P2/P4), and we have that P1 is a characteristic subgroup of

index p. Setting Po = P, we may form a characteristic series,



Po > FI > ... > P -1> Pn= lp,

such that |P/Pi/+1| = p for i E {0,..., n 1}. If H is a maximal subgroup of P, and

H 1 Pi, then H is also of maximal class. If P is of maximal class, then Z(P) is of order p.

Lastly, if 2 < i < n, then P/PF is also of maximal class.

Definition 5.0.4. We will define a family of groups that shall serve as a source of

examples for our later work. It has been shown by D.L. Johnson [22] that given a

commutative ring R with an identity, the set of formal power series,


00
Zaix, ai E R, a = 1,
i=1
together with the operation of substitution forms a group. We denote this group, G(R),

and refer to it as the Nottingham group over R. Johnson further demonstrated that the

set,


00
H,(R) = x + aix
i>n
is a normal subgroup of G(R). We will consider the quotients G,(R) = G(R)/H,(R),

where R = F,. We may think of the operation on G,(R) as being the usual operation

of substitution in the Nottingham group followed by truncation of all powers of degree

greater than n. We include a final result of Johnson that is of interest. Given a, E G(R),

where

a = x + aix + a2 + ... = x+ bixt + b2 + s ...









D.L. Johnson [22] calculated that,


[a, 0] = x + alb (m t) xm+t-1 + ... (5-1)

We note a similar formula holds in the quotient, G,(R).

We now record a theorem that describes a correspondence between certain p-groups

and Lie rings of nilpotence class < p 1. It will aid us in collecting a set of examples of

normally serially monomial p-groups. This is a result originally due to independent work

by Magnus [34] and Lazard [30], and we will refer to it as the Lazard correspondence.

Theorem 5.0.5. Given a nilpotent Lie ring, L, whose additive i/i ;'r is a p-/in'nr, and

such that the nilpotence class of L is < p 1, we ii;l 1f im,: a y'in]' GL on the same

i ;,i, i;1':,i' set of elements. To each nilpotent p-yin ';,nr, G, of class < p 1, we I,'ri ; I.fl,:

a Lie ring LG on the same underlying set. Moreover, these associations are inverse to one

another and have strong structure preserving properties as enumerated below.

(i) H is a -ni/i.i'n, of the i. 'nj' G if and only if LH is a subring of the ring La.

(ii) H is a normal -'ni. i',' of the p. 'nj' G if and only if LH is an ideal of the ring LG.

(iii) If H is a normal -'ni. i',' of the yi. n G (and thus, LH an ideal), the set [H, H]
coincides with the set [LH, LH].

(iv) The derived length of the yi. 'ny G equals the derived length of the ring LG.

(v) The nilpotence class of the yi. 'ny G equals the nilpotence class of the ring LG.
We now state preliminaries regarding the complex irreducible characters of a finite

group G.

Theorem 5.0.6. Suppose G = H x K. If x E Irr(G), then x = a* 3 where a C Irr(H) and

3E Irr(K).

We briefly discuss the definition of character induction and related properties.

Definition 5.0.7. Suppose H < G and 0 a complex character of H. We first define an

auxiliary function, 0' : G -i C in the following manner.












e0 0(g) g H
0 giH.

The character obtained by inducing 0 to G, denoted OG, may be expressed as,


(g) 00 (sgs1).
sEG

Remark 5.0.8. Suppose X1, X2 E Irr(H) and 0 = XI + X2. Using the above formula, one

may see that,
oG G G
S=Xl +X2.

We state without proof the following lemma regarding induction of characters through

a chain of subgroups.

Lemma 5.0.9. Suppose that X is a character of K, where K < H < G. Then, (Xy) G

XG

We also have a lemma that determines the kernel of an induced characters.

Lemma 5.0.10. Suppose that H < G and x is a character of H. Then,


ker X =- nEG (ker X)


Remark 5.0.11. The induction process described above may also be viewed in terms of

modules. Given H < G and 0 a complex character of H, let M be the corresponding

module. Then, one may show the module corresponding to 0", denoted MG, is given by

the tensor product M 0cH CG.

Definition 5.0.12. We ~i- that the group G is monomial if every irreducible character

of G may be induced from a linear character of a subgroup of G. We further that G is

no in Ill/;/ monomial if every irreducible character may be induced from a linear character

of a normal subgroup of G.









Theorem 5.0.13 (Frobenius Reciprocity). Suppose H < G, 0 is a character of H,

and X is a character of G. T1.':,,y [,]H and [,]G to be the respective inner products of

characters in H and G, and XH to be the restriction of x to the -;1l'p i'u H, we have,



[O",XHH [OG, G'

If H < G, the following lemma (stated without proof) allows us to discuss an action of

G on the characters of H, and on the set Irr(H).

Lemma 5.0.14. Suppose H < G and 0 is a class function of H. Then, for g E G, the

function 09 : H -+ C /. I;,. J by 09 (h) = 0(ghg-1) is said to be conjugate to 0. If

0 is a character of H, then so is 09, and if 0 is irreducible, so is 09. We further have
09192 = (091)92 and (1H) = 8(1H).

Definition 5.0.15. Suppose H < G and 0 c Irr(H). Then, the stabilizer of 0 in the above

action, is the subgroup,

Ic(O)= {g G 89= 0}.

It is called the inertia p '.;u of 0 in G.

Now, we record a key theorem regarding the restriction of irreducible characters to

normal subgroups.

Theorem 5.0.16 (Clifford's Theorem). Suppose H < G, X c Irr(G), and 0 is i/!;

irreducible constituent of the restriction XH. Then, if {01,... Ot} are the distinct G-

conjugates of 0, we have, XH = [XH,0] Z 1 Oi.

Remark 5.0.17. We note that in Clifford's Theorem, as {01,..., Ot} are the distinct

G-conjugates of 0, the orbit-stabilizer theorem yields t = 1.

We lastly mention an important divisibility condition of Ito on character degrees.

Theorem 5.0.18 (Ito's Theorem). Suppose A < G, and A is an abelian -';1 "uq'.. Then,

if c Irr(G), X() | [G : A].









CHAPTER 6
THE TAKETA PROBLEM

6.1 Derived Length vs. Number of Character Degrees in Certain p-groups

6.2 Introduction

Given a finite solvable group G, the relationship of the number of distinct character

degrees to the derived length of the group has been a subject of much interest. In 1930,

Ken Taketa [40] proved that for monomial groups, the derived length of the group,

denoted dl(G), is less than or equal to the size of the character degree set, denoted

|cd(G) Isaacs [20] and others have addressed the question, and it was conjectured that a

much stronger logarithmic bound exists. That is, there exist universal constants, C1 and

C2, such that the inequality,



dl(G) < Cilog(|cd(G)|) + C2

holds for all finite solvable groups G. In a series of papers, Thomas Keller ([24], [25], [26])

effectively reduced the problem to the case of finite p-groups. Keller [27] then considered

the case of normally monomial p-groups, P, of maximal class. In this setting, he proved

the inequality,


1 11
dl(P) < cd(P) + -. (6-1)
2 2

For a normally monomial p-group, P, of maximal class, there is a characteristic series,



P P > Pi > ... > P.- 1 > P.,

such that for X c Irr(P) there exists i E {0,... n} and A E Irr(Pi), where A is linear and

AP = X. We investigate here the case of p-groups for which every irreducible character

may be induced from a normal series, but the group itself need not be of maximal class.

We define these groups to be no i/br/ll;i 1. i,ll; monomial p-groups. We also examine

interesting character theoretic properties satisfied by normally serially monomial p-groups.









6.3 Normally Serially Monomial p-groups

Definition 6.3.1. We define a p-group, P, to be no, 'ill;i ,. i'll- ; monomial, if there is a

normal series of subgroups,



P = P> P > ... > P- I > P.= Ip,

such that Pi/P| = p for i E 1,..., n, and for every x E Irr(P), there exists Pi with

0 < i < n and A C Irr(PF), such that A(1) = 1 and AP = X. We call the series of subgroups

a normal series of induction.

Example 6.3.2. Suppose that P is a normally monomial p-group of maximal class.

We will show that P is normally serially monomial. Specifically, we show that a normal

series of induction is the standard characteristic series of any p-group of maximal class.

We first consider any irreducible character x such that x(1) > p2. This implies that

X may be induced from a normal subgroup of index at least p2. As mentioned in the
preliminaries, in a p-group of maximal class, each normal subgroup of index greater

than or equal to p2 belongs to the lower central series of the group. This leaves the

irreducible characters of degree p. Let X C Irr(P) such that x(1) = p. Assume that x is

induced from H < P where H is maximal but H / P1. Then, H is of maximal class, so

[P : H'] = [P : H] [H : H'] = p3. Further, as H' is characteristic in H, and H < P, H' < P

and so, H' = P3. This implies that ker X > P3 > P,. Viewing x as an irreducible character

of the quotient P/P,, we have that Xp, is a sum of linear characters. Hence, x is induced

from a linear character of PI/P' in the quotient P/P{, and therefore is induced from a

linear character of Pi in P. As the selection of x and H were arbitrary, every irreducible

character of degree p is induced from the subgroup P1. So, the characteristic series of P is

also a normal series of induction.

We now give a concrete example of a normally monomial p-group of maximal class.

Example 6.3.3. We will consider appropriate quotients of the Nottingham group over

Fp for p a prime. In his 1990 thesis, I. York [41] shows that the groups G, (Fp), with









4 < n < p + 2, are normally monomial, of maximal class, and have derived length

[log2 n]. By the above, this establishes a family of normally serially monomial p-groups. It
also establishes that the derived length of normally serially monomial p-groups may not

be bounded independent of the prime p. It is worthwhile to note that a subset of these

examples was initially found by L.G. Kovacs and C.R. Leedham-Green [29] through a

different method. One may view their approach as taking the n-dimensional vector space

L(n) = (e1,...,en), 3 < n < p over Fp and imparting a Lie structure according to the

following relations.



[e, ej] = (
0 i +j > n.

If we denote Li(n) = (e, ..., en), we have the lower central series for L,


LIn) >L3n) > ...>L, n) > 0,

and L(n) is of maximal class. Then, .i the image of L(n) under the Lazard correspondence,

is also of maximal class. In his thesis, York [41] showed that G,i(Fp) [ i. for

3
Example 6.3.4. We consider a normally monomial p-group of maximal class of

isomorphism type different from those in Example (6.3.3). The group is due to Keller

[27]. We take L = (e1, e2, ..., e) to be an 8 dimensional vector space over Fp with p > 8.

We turn L into a Lie algebra with the following relations. [el, e] = ei+ for 2 < i < 7,

[e2, 63 = C7, [e3, 64 = 67, [e3, 65 = [2, 64 = 8, [2, 65 = -7, [2, 61] = -2es, and

[ei, j] = 0 if not explictly listed. Keller's example is the Lazard correspondent of this Lie
algebra.

We now consider a family of normally serially monomial p-groups which are not of

maximal class.









Example 6.3.5. Let p be a prime, p > 2, and fix q = pr where r > 2. We show the

Nottingham quotients, G = G, (Fq), are normally serially monomial for n < p + 2. For

1 < j < n, set K,(j) = Hj(F)/H,n(Fq). We show each irreducible character may be

induced from a subgroup belong to the normal series,


G, = K(1)) > K.(2) > ... > K,(n 1) > K,(n) = 1tG.

Therefore, the above series may be refined to a normal series of induction. The argument

uses an approach that York [41] employ -1 to calculate the character degree set of GQ. We

proceed by induction on n. For n = 2 and n = 3, G, is abelian and the result holds.

Now, take X c Irr(G,). Suppose kerx > Z(G,). Then, we may view X as an irreducible

character of G,/Z(G,) G,_1. So, by the induction hypothesis x is induced from

K,(b)/Z(G,), 2 < b < n 1 when viewed as a character the quotient. We conclude that x
is induced from K,(b) when viewed as a character of the original group. Now, we consider

the case when X c Irr(G,) where kerX Z Z(G,). Set A = K,([] ). We may see from

equation (5-1) that A is an abelian normal subgroup. Consider an irreducible constituent,

A, of XA. We note, A is linear as A is abelian. Further, York demonstrated the inertia

group of A is A, and hence A induces irreducibly to G,. Then, by Frobenius reciprocity,

0 / [A, XAIA = [AG,' X] G Hence, A0" = X and every character may be induced from a
linear character of the given normal series. Now, we address the class of G,. In an article

of R. Camina [7], it is shown that,


[H(), H( H (F,,) if i j mod p

Hi+j (F,,) if i mod p.

This leads to a similar calculation in the quotient G,. We have,

K,(i + j) if i mod p, i+ j
[K(i, K(j)] = K(i + j + 1) if i mod p, i + j < n 1 (6-3)

lGQ if i + j > n.









This establishes that [G,, G,] = K(3), and as n < p + 2, [G,, K(i)] K"(i + 1) for

i E {2,... n 1}. So, the class of G, is equal to n 2. However, as we may consider

G, = { x + a2x2 + ...+ ax ai E F, |G, = p pn-2 r(f-2). So, G, is not of
maximal class.

We now describe some of the interesting properties held by normally serially

monomial p-groups.

Theorem 6.3.6. Suppose P is a noc mialh.l ,.i',ll; monomial p-_ipj;,ir, with normal series

of induction,


P = P> P > ... > P.-I > P.= Ip.

Then, for i E {0,...,n}, each Pi is a no inll, j ,.
series of induction,


Pi> > Pi+> .. P-,_1 >P,= Ip.

Proof. Let i E {0,..., n} and X e Irr (Pt) such that X(lp) / 1. Let 0 e Irr(P) such

that [Op,, Xlp. / 0. As P is normally serially monomial, there exists j E {0,..., n} and

A e Irr(Pj), A(lp) 1, such that AP 0. As (A ) 0 we have Ap e Irr(P). By

Frobenius reciprocity, [Op,, A] i/ 0. Now, if A = 71 +72 for 71, 72 characters of Pf, then

(A ) = 7P + 72, and (A )< is a reducible character of P. However, (Ap) = AP 0,

and 0 e Irr(P). So, A^ e Irr(Pi). As Pi < P, Clifford's Theorem states that Op, is a

sum of P-conjugate irreducible characters, each of the same degree. With x and A^ each

irreducible constituents of Opz, we have X(1p) = A^(lp). Now, as AP = 0, Frobenius

reciprocity yields that [0pj, A] / 0. Again by Clifford's theorem, Op. is a sum of

irreducible characters of the same degree, so Op. is a sum of linear characters. Then, with

(O)p = Op, and X is a constituent of Opt, Xp, is a sum of linear characters. Let a E Irr(P,)
such that a(lp) 1= and [Xp,, a] / 0. By Frobenius reciprocity, [a x, X] / 0. So, ap

is a sum of irreducible characters one of which is X. Recalling the formula for character









induction, we have, a^p(lp) = Ap(lp) = X(lp). This yields ai = x and X is

induced from a linear character of Pj.

Theorem 6.3.7. Suppose P is a no ail.ll/ ,-.'ll;' monomial p-/j *;,'j', with normal series

of induction,



P = > Pi > ... > P- > P= Ip.

Then, for i = 1,..., n, P/P = P is a noi mlrll ,.;'ll.; monomial p-ji'n.;,I with normal

series of induction,



P>P > ...> P1 > P =1p.

Proof. Let i C {0,..., n} and y c Irr(P). Then, 7 may be inflated to an irreducible

character X c Irr(P), where ker X > Pi. As P is normally serially monomial, there exists

1 < j < i and a linear character, AE Irr(Pj), such that AP = X. As ker X > P,, ker A > Pi,

so we consider A C Irr(P,). Let gPi E P. We have,




y(gPi) = X(g) = AP(g)

L A(xgx-1)
J JEP


Pi
zPEP

L (xgx- ,P i)


3 P(gP).

-P
So, A c Irr (Ps), and ~ is induced from a linear character of P,.

Next, we show that any nonlinear character of any subgroup in the normal series of

induction will, in fact, induce irreducibly to the entire group.









Theorem 6.3.8. Suppose P is a noi ail.J ,.ill1, monomial p-i/-;,'rj, with normal series

of induction,


P = PO > Pi > ... > P- > P Ip.

Let i E {0,..., n} and X E Irr (Pi) such that X(1) / 1. Let 0 c Irr(P) such that

[p, X]p, O0. Then, XP = 0 G Irr(P).

Proof. As P is normally serially monomial, there exists j c {0,..., n} and A E Irr(Pj)

such that A(lp) = 1 and AP = 0. As P, <, P, [Op\, A] / 0, Clifford's Theorem implies

that Op. is a sum of linear characters. So, j > i. Now, (Ai) 0, so A E cE Irr(Pi).
Then, by Frobenius reciprocity, [6p, Ai] / 0, that is, Ap is an irreducible constituent

of Op,. By hypothesis, X is an irreducible constituent of Opt, so Clifford's Theorem yields

that A^(lp) -= (lp). This implies that XP(lp) = (A)p (1p) = 0(1p). By Frobenius

reciprocity, [X, 0] p / 0. So, XP is a sum of irreducible characters, one of which is 0, and

noting XP(lp) = 0(lp), we have XP = 0.

Corollary 6.3.9. Suppose P is a no 'i ni'l ,.;'ll.; monomial p-_i/;'ni, with normal series

of induction,


P = PO > PI > ... P- > P.= Ip.

Let i c {0,..., n} and X e Irr (Pi) such that X(1) / 1. Then, the inertia ji"-q'; of X, I(X),

is Pi.

Proof. By Theorem (6.3.8), XP = 0 c Irr(P). So, by Frobenius reciprocity,


1 p, ]- [Opp, x1i.









Then, Clifford's Theorem yields Op, = Y- Xi where {i = ,... Xt} are the distinct
P-conjugates of x and t -= IPI. Now,

IPW
XP(1p) X(lp) -= (lp) = Op(lp) Yx(1p). (6-4)
1 I i= 1

As conjugate characters have the same degree, (6-4) implies,

|PI |PI
x(lp) = J(lp).

So, as Ip(X) > Pi, Ip(x) = Pi.

We now show that the character degree set of the entire group determines the
character degree set of every subgroup in the normal series of induction.
Theorem 6.3.10. Suppose P is a no mall;l /. !.ll;i monomial p-,'j' ;, with normal series

of induction,


P = PO > Pi > ... > P.-1 > P.= Ip.

Suppose P has set of character degrees, cd(P) = {I,pa",pa2,. .,pam }. Then, for i E

{0, n},
cd(Pi) t, pa,-i, pa-i.,.. pa--i ,

where for j e {1,..., m}, pa}-i = paj-i if aj i > 0, and paj-i = 1 otherwise.

Proof. Let X c Irr(Pi) such that X(1p) / 1. By Theorem (6.3.8), XP G Irr(P). Now,

XP(1p) P : P| (p), so P : P| (p) paj for some j {1, m}. Clearly, aj > i,
and we have, X(Ip) -= jI pa- So, cd(P,) C {1, pa -i, pa2- i... pa-i Now,

let 0 e Irr(P) such that 0(lp) = paj where aj > i. Then, as P is normally serially
monomial, there exists A E Irr(Pk) such that A(1p) = 1 and AP = 0. As AP(lp) =

P: Pk A(lp) (lp) pa, |P :k paj, and a =- k. Now, as A' = (A p)P 0 and

0 c Irr(P), A c Irr(Pi). Moreover, AP (p) Pi : Paj A(lp) : paj So,

cd(P,) -1 pa r-i pa2-i. pa--i.









This establishes the following corollary regarding abelian subgroups of maximal order.

Corollary 6.3.11. Suppose P is a no malbil ,.ill.; monomial p-j,.;,I' with normal series

of induction,



P = P> P > ... > P- I > P.= Ip.

Suppose P has set of character degrees, cd(P) = {1, p",p2,... ,p }m) where ai+ > ai for

i {1,... ,m 1}. Then, Pa, is an abelian .-;,1,' i.' of maximal order.

Proof. By Theorem (6.3.10), cd(Paj) = {1, pal-a, pa2-a,..., pam-a, } where pj-a -

paj-am if aj am > 0, and paj--m = 1 otherwise. So, cd(Pa,) = {1}, and Pa. is abelian.
Let H be any abelian subgroup of P. Let X c Irr(P) such that X(lp) = pam and A c

Irr(H) such that [AP, x] / 0. Then, AP(lp) P : H A(lp) > a(lp) = p.a P: PaJ

As A(lp) = 1, the result follows.

We now give a theorem which determines the number of characters of a particular

degree in terms of the group theoretic invariants. This generalizes a result of Thomas

Keller [27] which gave a group theoretic determination of the character degrees (without

multiplicity) of normally monomial groups of maximal class.

Theorem 6.3.12. Suppose P is a nociall;.'i ,.;'ll.; monomial p-, .;'j', with normal series

of induction,


P =P> PI>- .. > P-1 > P = Ip.

Then, for i e {1,..., n}, |{X e Irr(P) X(lp) }- } [P:PJI -[ .

Proof. Let 0 c Irr(P) such that 0(1p) = p'. So, 0 is induced from a linear character,

A e Irr(Pi). As Pj < P, and Pf characteristic in Pi, P> < P. As ker A > P1,


ker 0 = nEp (ker A)g > P,'. (6-5)









Now, suppose ker6 > P-1. Set P = P/P7 i, and view 0 e Irr(P). We note that Pi- is
an abelian normal subgroup of P. Ito's theorem then yields that 0(1p) = pi : i- =

pi-1. This is a contradiction, so,
ker 0 P1__ (6-6)

Set A {e Irr(P/P) | kery > P1_ PJ. In

consideration of equation (6-5) and equation (6-6), we have that A = { E Irr(P/P/) X| (lp) < pi}
and B = {X Irr(P/Pi') X(lp) = p'}. We have,



IP/ E X(1)2
XEIrr(P/P/)

SX(1p)2 X(p)2
XEA XEB

SP/Pf-+ +p- *B

= |P/Pfi + p: p 2. B1.



Solving for |B|, we get,

|P: P' I P: P
| pp p pi




|P : P,


IP : Pil

|P:Pi









Before using Theorem (6.3.12) to give a group theoretic description of |cd(P)|, we

record two results of a similar spirit to Theorem (6.3.12).

Theorem 6.3.13. Suppose P is a noc lmlbli/ monomial p-'ip';,r of maximal class, with

normal series of induction,



P = PO > Pi > ... > P- 1 > P.= Ip.

Suppose 1 < k < n. Then, there is a unique i C {1,..., n} such that Pi_, > Pk > P*.

Further, if 3 < k < n and D = {X c Irr(P) | ker = Pk}, IDI = p-(-1), with i as above.

Proof. For j 0, .., n 1}, P' is characteristic in Pj and PI
unique normal subgroup of index pr for r > 2, we know P' = P, for some s C {2,..., n}.

Since P,' > P,,, for 1 < k < n, there is a unique i {1,..., n} such that Pl_, > Pk >- P.

Now, suppose 3 < k < n and i as before. Let 0 c D. It follows that 0(1p) > 1.

As P is normally monomial of maximal class, P is normally serially monomial. Hence,

equation (6-5) and equation (6-6) in Theorem (6.3.12) establish that 0(lp) = p'. We set

C = {0 e Irr(P) | ker 0 > Pk-1k}. We note, as irreducible character kernels are normal, and

P has a unique normal subgroup of index pr for r > 2, we may view Irr(P/Pk) = C U D.

Similar to the argument in Theorem (6.3.12), we have,


\P/Pkl x(Ip)2 + X(p)2 Pk-1 +2"i | .
XEC XED



Solving for |D|, we get,

D P : PkP P: P-1
p2i
P/Pk-1 (P )
P2i
p-1 (p t1)
P2i











The above theorem regarding the kernels of irreducible characters allows us to

establish a similar corollary regarding centers of irreducible characters.

Corollary 6.3.14. Suppose P is a rn. ,I,,ill; monomial p-.pI .;,Ir of maximal class, with

normal series of induction,



P = P> P > ... > P- I > P.= Ip.

Suppose 3 < k < n and i {1,...,n} such that Pi1 > Pk > P[. Then, {X e Irr(P) | Z(X) Pk-1}

{X c Irr(P) | ker = Pk}. In particular,

pk-1 I
\{x c Irr(P) | Z(X) = Pk-}| (p )

Proof. We first recall that for j > 2, P/Pj is of maximal class and the center of any

p-group of maximal class is of order p. Now, take 0 c Irr(P) such that ker 0 = Pk

(existence is guaranteed by Theorem (6.3.13)). By Lemma 2.27 in [21], Z(O)/kerO -
Z(P/Pk). Then, |Z(O)/ker6| p and Z(O) < P. Finally, as P has a unique normal

subgroup of index pr for r > 2, we have that ker 6 = Pk if and only if Z(O) = Pk- 1. This

implies that,

\{x G Irr(P) | Z(X) = Pk-1} (p )

The final equality is by Theorem (6.3.13).

As a corollary of Theorem (6.3.12), we obtain the following result.

Corollary 6.3.15. Suppose P is a nc inll;l .I ,.;'ll.; monomial p-_i/.;'r, with normal

series of induction,


P = PO > P > ... > P- > P Ip.

Then,











cd(P)- I {|P: P, i I{l,...,n-1} andP', > P,'}.

Proof. Take i > 1. By Theorem (6.3.12), |{ e Irr(P) X(1p) = pi}| > 0 if and only if

P<-1 > P,. This gives the desired result.

We can now determine the size of the character degree set of a normally serially

monomial p-group in group theoretic terms.

Corollary 6.3.16. Suppose P is a nc m-ll;l .I ,.!ll;i monomial p-ip.;'-r, with normal

series of induction,



P = PO > P > ..I > P-I > P= Ip.

Then, cd(G) |{Pj e {0,...,n}}|.

With an additional hypothesis, one can show Corollary (6.3.15) provides a logarithmic

bound between the derived length and the size of the character degree set.

Corollary 6.3.17. Suppose P is a nc nill;l ii ,.;'ll; monomial p-i'u.;'r, with normal

series of induction,


P = PO > Pi > ... > P- > P p.

Suppose there exists b cE Z such that for i E {0,..., n} and P' / Ip, we have P' > P>+b

Then,


dl(P) < 1 + log2 (b cd(P) b + 2).

Proof. Enumerate cd(P) = {po = l,pa ",... pa"} where aj+i > ai for i e {0,... ,m}.

Consider pak,pak+l e cd(P). According to Theorem (6.3.15), Pa k P +1

Pak1-1 > Pa k. By hypothesis, ak+l ak < b. Now,

am ao = (a ao) + (a2 a) + ... + (am am-).









So, am < b m + 1 = b (|cd(G)| 1) + 1. Further, A. Mann [35] has proven that for a

normally monomial p-group, G, with maximal character degree p',


dl(G) < 1 + log2 (r + 1) .


(6-7)


Then, equation (6-7) and the bound on am yield,


dl(P) < 1+ log2 (b m + 2) 1+ log2 (b |cd(P) b + 2).




Example 6.3.18. The approach of Corollary (6.3.17) may be applied to the family of

groups mentioned in Example (6.3.3). Fix a prime p, p > 2. We consider the Lie algebra

L = (e, ... ep), and note that if we denote Li = (e, ... ep) and Lp I = 0, the lower

central series of L is given by,

L = L, > L3 > ... > Lp > Lp+I 0.


The Lie relations,


[ei, ej] (i


j)ei+j i+j 0 i+j>p


allow us to calculate that,

{ L2i+1 2i + 1 < p
0 2i + 1 > p.

So, L 1_, > L' for i E {2,..., 1}. Now, we take P as the Lazard correspondent, with

characteristic series,


S= Po > P > ... > P-1 > = Ip.

According to the structure preserving properties of the Lazard correspondence, P1_, > P'

for i G {1,..., 2'}. Then, by Theorem (6.3.15), cd(P) ,p,p2,... ,p We may

apply Corollary (6.3.17) with b = 1. This results in the bound,











dl(P) < 1 + log2 (|cd(P)| + 1) 1 + log2 (1 +

The Lie relations establish that L(k) = L2k+ _- for k such that 2 k+1

derived length of L is then the minimal integer mo such that 2"m+1


log2 (p + 3).

- 1 < p + 1. The

- 1 > p. This gives,


dl(L) = dl(P) = [log2 (p + 1)]

For the family of groups in this example, our bound is close to the actual value.

Example 6.3.19. We may also apply Corollary (6.3.17) in the case of the Nottingham

quotients in Example (6.3.5). Fix p > 2 and take q = pr with r > 1. As in Example

(6.3.5), we consider n < p + 2 and Gn = Gn (F,). For 1 < j < n, set K,(j) =

Hj(Fq)/Hn,(Fq) and form the normal series,

G, = K(1)) > K.(2) > ... > K,(n 1) > K,(n) = 1tG.

It was shown in Example (6.3.5) that each irreducible character may be induced from a

subgroup of the above series. To meet our definition of a normal series of induction, we

refine the above series to a series where successive quotients have order p. We obtain,


G, = K(1) > ... > P, = K,(2) > ... ... > P(,_2) = K(n 1) > ... > K,(n) = t1 .


According to the equations (6-3) in Example (6.3.5), we have


K' (t)


(6-8)


K(2t + 1) if 2t + 1 < n

1G if 2t + 1 > n


So, for t c {1,... n} such that 2t + 1 < n, we have K' t) / K' t + 1). It is clear we may

apply Corollary (6.3.17) with b = 2r, and in fact, it may be shown we can that we can
take b = r. This gives the bound,


(6-9)


dl(G,) < 1 + log2 (r (|cd(G,)| 1) + 2).









York [41] has determined that |cd(G,)|


[l] which then gives,


dl(G) < 1 + log2 r r + 2) (6-10)

Similar to the case in Example (6.3.18), equations (6-8) yield that G ) K,(2m+l 1)
for m such that 2m+1 1 < n. Hence, the derived length is the minimal integer mo such
that 20+1 1 > n 1. We obtain,


dl(G,) = [log,2 n]

It is evident that our bound in equation (6-10) is most accurate for small values of r.









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BIOGRAPHICAL SKETCH

Tim Bonner was born in Trenton, New Jersey. A northerner until the age of eighteen,

he obtained a Bachelor of Science in chemical physics from Rice University in 2001. After

a short stay in the Dallas business world, Tim taught high school algebra for a year at

Westside High School in Houston, and then returned to the formal study of mathematics

at the University of Florida in 2003. He completed his Ph.D. in the area of finite group

theory in 2009.





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Iamsincerelygratefultomyadviser,AlexandreTurull.Withunyieldingpatienceandconstantsupport,hehasbeenintegralinmyacademicandpersonalgrowth.Ifeelindebtedtohavebeenhisstudent,andIamfortunatetoknowsuchaninspiringindividual.Ialsowillforeverappreciatetheencouragementofmyfamily.Overthepastsixyears,myparentsandsisterhaverespondedtoeachstepforwardandeverysetbackwithcalmassurance.Finally,mywife,Emily,haswitnesseditallonlyaglanceaway,andthisworkundeniablybearsthesteadystrengthofherhand. 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 5 ABSTRACT ........................................ 6 CHAPTER 1INTRODUCTION .................................. 8 1.1ProductsofCommutators ........................... 8 1.2FurtherGenerationProblems ......................... 9 1.2.1ProductsofConjugacyClasses ..................... 9 1.2.2ProductsofCharacters ......................... 10 1.3TheTaketaProblem .............................. 11 2MATHEMATICALPRELIMINARIESFORCHAPTERS3AND4 ....... 13 3PRODUCTSOFCOMMUTATORSANDBARDAKOV'SCONJECTURE ... 17 3.1ACharacterIdentityofBurnside ....................... 18 3.2AnalysisandResults .............................. 21 3.3Bardakov'sConjecture ............................. 26 4PRODUCTSOFCONJUGACYCLASSESANDCHARACTERS ........ 28 4.1ConjugacyClassCoveringNumbers ...................... 28 4.2ComparisonwithPreviousBounds ...................... 33 4.3CharacterCoveringNumbers ......................... 36 4.4ComparisonwithPreviousBounds ...................... 39 5MATHEMATICALPRELIMINARIESFORCHAPTER6 ............ 43 6THETAKETAPROBLEM ............................. 48 6.1DerivedLengthvs.NumberofCharacterDegreesinCertainp-groups ... 48 6.2Introduction ................................... 48 6.3NormallySeriallyMonomialp-groups ..................... 49 REFERENCES ....................................... 64 BIOGRAPHICALSKETCH ................................ 67 4

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Table page 4-1BoundComparisonforcn(C)forA5 34 4-2BoundComparisonforcn(C)forSL2(5) ...................... 35 4-3BoundComparisonforcn(C)forSL2(8) ...................... 35 4-4BoundComparisonforcn(C)forPerfectGroupofOrder1080 .......... 35 4-5BoundComparisonforcn(C)forM11 36 4-6BoundComparisonforccn()forA5 40 4-7BoundComparisonforccn()forSL2(8) ...................... 41 4-8BoundComparisonforccn()forPSU3(3) ..................... 41 4-9BoundComparisonforccn()forM11 41 5

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LetGbeanitegroup.Itiswell-knownthattheelementsofthecommutatorsubgroupmustbeproductsofcommutators,butneednotthemselvesbecommutators.Anaturalquestionistodeterminetheminimalinteger,(G),suchthateachelementofthecommutatorsubgroupmayberepresentedasaproductof(G)commutators.AnanalysisofaknowncharacteridentityallowsustoimprovetheexistinglowerboundsforjGjintermsof(G).Thetechniqueswedevelopalsoaddresstherelatedfollowingquestion.SupposewehaveaconjugacyclassCofanitegroupGsuchthathCi=G=G0.Onemayaskfortheminimalintegercn(C)suchthateachelementofGmaybeexpressedasaproductofcn(C)elementsoftheconjugacyclass.Again,weimprovetheknownupperbounds,thistimeforcn(C). Oursecondfocusistherelationbetweenthederivedlengthofanitesolvablegroupandthecardinalityofthesetofcharacterdegreesinthesamegroup.Overthepastfewdecades,thistopichasbeenexploredbyIsaacs,Gluck,Slattery,andmostrecently,byThomasKeller.ThereisastandingconjecturethatuniversalconstantsC1andC2existsuchthatforanynitesolvablegroupG,dl(G)C1logjcd(G)j+C2: 6

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7

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13 ],thatDedekindwasthersttointroducetheideaofacommutator,anelementofthegroupoftheforma1b1ab,fora;b2G.Dedekindalsoinitiatedthestudyofthesubgroupgeneratedbythesetofcommutators,latertobecalledthecommutatorsubgroup.Itwassoonrecognizedthateachelementofthecommutatorsubgroupneednotbeacommutator,andsuchanelementissaidtobeanoncommutator.WilliamBenjaminFite[ 12 ]wasthersttopublishanexampleofagroupcontainingsuchanelement,thoughheattributedtheexampletoG.A.Miller.WilliamBurnside[ 5 ],in1903,subsequentlydevelopedacriteriontodeterminewhetheranelementofthecommutatorsubgroupwasindeedacommutator.Heshowedthatg2Gwasacommutatorifandonlyif whereIrr(G)denotesthesetofirreduciblecomplexcharactersofG. Thisidentitycanbeapowerfultoolindeterminingtheexistenceofnoncommutatorsandweshallmakesignicantuseofageneralizationof( 1{1 ).Thequestionthenarisesastoidentifyingoftheminimalinteger,(G),suchthateveryelementofthecommutatorsubgroupmaybewrittenasaproductof(G)commutators.Thisinvariant(G)hasbeenasourceofconsistentinvestigationthroughoutthe20thand21stcenturies.Inthe1960'sPatrickGallagher[ 15 ]determinedaninequalitybetweenthesizeof(G)andtheorderofanitegroup,jGj.Later,in1982,RobertGuralnick[ 18 ]demonstratedthatforanypositiveintegern,onemayconstructanitegroupGsuchthat(G)=n.HealsodeterminedtheminimalnitegroupsG,withrespecttoorder,suchthat(G)6=1[ 19 ].ThefamousconjectureofOysteinOre[ 38 ],statingthat(G)=1foranynitesimple 8

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17 ],EllersandGordeev[ 11 ],andLiebeck,O'Brien,Shalev,andTiep[ 32 ].Ourworkregarding(G)returnstotheearlierconsiderationsofGallagher.WeimprovetheknownlowerboundforjGjforagivenvalueof(G)andobtainthefollowingasourrstmainresult. 15 ]byafactorof2.Moreover,weusethisresulttoconrmandstrengthenaconjectureofV.G.Bardakov( 3.0.18 )posedinthemostrecenteditionoftheKourovkaNotebook[ 36 ].Precisely,weobtainthefollowingtheorem. 250: 23 ]haveshownthat(G)2f1;2gforallgroups,G,suchthatjGj1000.Ourresultsregardingproductsofcommutatorshavebeenpublishedina2008volumeoftheJournalofAlgebra[ 4 ]. 1.2.1ProductsofConjugacyClasses 1 ]in1985thattheexistenceofanintegernsuchthatCn=GwasequivalenttotheconjugacyclassgeneratingG, 9

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33 ].WewillconsiderthegeneralcaseofGperfect,withtheminimalintegercalledtheconjugacyclasscoveringnumberanddenotedcn(C).Intheirwork,Arad,Stavi,andHerzog[ 1 ]obtainedavarietyofupperboundsontheconjugacyclasscoveringnumber.Towardsthelatterendofthe1990'sandintotherstpartofthecurrentdecade,DavidChillag([ 9 ],[ 10 ])producedalternateupperboundsforcn(C).Inmanycases,weimprovetheknownupperboundsoftheconjugacyclasscoveringnumber,ourmainresultbeingthetheorembelow. 8 ]showedtheexistenceofccn()ifandonlyifZ()=1G.Further,aswithcn(C)onemaydetermineupperboundsontheinvariantccn().In1989,ZviAradandHinnitLipman-Gutweter[ 2 ]obtainedboundssimilartothoseofArad,Stavi,andHerzogregardingconjugacyclasscoveringnumbers.DavidChillag([ 9 ],[ 10 ])alsoobtainedboundsforccn()whichparalleledhisworkwithconjugacyclasscoveringnumbers.Weagain,inmanycases,improvetheknownupperboundsonccn(),byananalysisofacharacteridentity.Ourmaintheoremisthefollowing. 10

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40 ]provedthatformonomialgroups,thederivedlengthofthegroup,denoteddl(G),islessthanorequaltothesizeofthecharacterdegreeset,denotedjcd(G)j.Itwasconjecturedthatastronger,logarithmicboundheldforallsolvablegroups.Isaacs[ 20 ]rstproveddl(G)3jcd(G)j2foranysolvablegroupGin1975.OneyearlaterT.R.Berger[ 3 ]proveddl(G)jcd(G)jinthecasejGjisodd,andthenin1985,DavidGluck[ 16 ]showeddl(G)2jcd(G)jinthegeneralsolvablecase.Towardstheendofthe1990's,ThomasKellerpickeduptheproblemandinaseriesofpapers([ 24 ],[ 25 ],[ 26 ]),hereducedtheconjecturetoaconsiderationofnitep-groups.Slattery[ 39 ]hasshowndl(P)jcd(P)j1forap-group,P,suchthatcd(P)=f1;p;p2;:::;pngandMoreto[ 37 ]improvedSlattery'sresult,establishingalogarithmicboundintermsoftheexponentofthemaximalcharacterdegree.In2004,Keller[ 27 ]consideredthecaseofnormallymonomialp-groups,P,ofmaximalclass,andinthissetting,heproveddl(P)1 2jcd(P)j+11 2. 11

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27 ]inwhichhedeterminesthesetofcharacterdegrees(butnotthemultiplicity)ofanormallymonomialp-groupofmaximalclass.Withanadditionalhypothesis,onemayusetheabovetheoremtoobtainalogarithmicbound. dl(P)1+log2(bjcd(P)jb+2): 12

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Wenowgivesomebackgroundforthemathematicstoappearinthenexttwochapters.ThroughoutwetakeGtobeanitegroup. 13

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14

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Wenowrecordsomerelationsbetweenthecharactersthatprovehelpfulinmanysituations,includingourcalculationswithcommutators.Welistthemwithoutproof. 15

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16

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WenowaddressBardakov'sconjectureandboundsontheinvariant(G). AresultofRobertGuralnick[ 19 ]statesthattheminimalorderofagroupwithrespecttothepropertythat(G)6=1is96.Thereareexactlytwoisomorphismtypeswiththispropertyandofthisorder.Oneofthetwoisgivenby, 18 ]furtherconstructed,foranyintegern,anitegroupsuchthat(G)=n.Theexampleispresentedbelow. WenotethatinGuralnick'sconstruction,thesizeofthegroupincreaseswithincreasingvaluesof(G).Itisnaturaltoconsidertherelationof(G)withjGjandour 17

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36 ],Bardakovsuggestedthefollowing. 6,andtheinequalityisattainedonlyatthesymmetricgroupon3letters. WeproveastrengthenedversionofBardakov'sconjecture.Ourproofusescharactertheoreticconsiderations,andlargelycontinuesapathoutlinedbyWilliamBurnsideandPatrickGallagher. Wenotethehistoryoftheproofoftheabovelemma.Thecasen=1wasprovenbyWilliamBurnside[ 6 ]inhistext`TheTheoryofFiniteGroups,'andthecasen=2wasposedasanexcercise.In1962,thegeneralcasewasformallyprovenbyPatrickGallagher[ 14 ]. 14 ]in1962,however,wenotethatGallagherproveditinthemoregeneralsenseofcompact(possiblyinnite)topologicalgroupswithintegralsusingHaarmeasureasopposedtonitesums.Webeginwithtwolemmas.

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1 jKj jKj!()I=1 1

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1 Wecompletetheproofof( 3{1 )byprovingamoregeneraltheorem.WecountthenumberofwaysanelementginGmayberepresentedasaproductofncommutators.Wenotegmayhavemultiplerepresentationsasaproductofncommutators.Forexample,givenapositiveintegern,theidentityhasmultiplerepresentationsasaproductofncommutators. 3{1 ). 20

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3.1.3 );=jGj2n1 15 ]andwestateithereinawaymoreconvenientforourpurposes.

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Setting jGjX2Irr(G);(1)=fi(1)(g); 3{2 )as, Takeq(x)=rQj=1(xj).Now,0=1,andi1 4fori1,implythatq(0)6=0whileq(i)=0for1ir.This,togetherwithequation( 3{3 ),impliesthatn0,asr>n.By( 3{3 ),weknowthat, Wehave, 22

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jG0jrXi=1jaijjGj jG0jrXi=1aip(i) jG0jp(1) 3{4 )=jGj jG0jmp: jG0j=jGj jG0j(jG0j1)
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Weadopttheconventionthat( 3{6 )holdsifp(j)=0.Wehave,q(1) 21 21c c: 3{6 )impliesthatthelemmaholds. Proof. 3.2.2 )wemayndp(x)2R[x]ofdegreen,suchthatp(1)6=0and,p(1) 3.2.1 )givestheresult.

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Proof. 2n+2 2nYi=1(f2i1)1>nYi=1f2i;(3{7) andtheresultfollowsfromCorollary( 3.2.3 ). ForanynitenonabeliangroupG,Gallagher[ 15 ]showedthatjG0j1=2((G)+1)!((G)1)!+1: Proof. 3.2.1 )showstheGhasatleastn+1distinctcharacterdegrees.Further,Corollary( 3.2.3 )impliesthat, 25

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Proof. 3.2.1 )showsthatGhasatleastn+1distinctcharacterdegrees.Then,Corollary( 3.2.4 )impliesthat,pa=jG0j>n1Yi=1f2in1Yi=1p2i=pn(n1); Proof. 3.2.1 ),(G)2.Letp(x)bethelinethrough(1+2 3.2.1 )givestheresult. 3.2.1 ).WenotethereexistsagroupGsuchthatjGj=128,thesetofcharacterdegreesisf1;2;4g,andG0containsanoncommutator.Forthisgroup,Corollary( 3.2.6 )givesjG0j8,Corollary( 3.2.7 )givesjG0j10,andtheactualvalueofjG0jis16. 6withtheboundobtainedonlyatthegroupS3.Further,limjGj!1(G) 250.

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3.2.5 ),wehave, ((G)+1)!((G)1)!=1 ((G)+1)((G)1)!2:(3{8) So,(G) 6if(G)3.As(G)=1forallgroupsGwherejGj12,therststatementfollows.Weletc.d.(G)representthesetofcharacterdegreesofG.Then,Theorem( 3.2.1 )gives,(G) jGjp jGj=1 250.Then(G)>4.By( 3{8 ),(G) 64!2=1 3456,andthelaststatementresults. Proof. 3.3.1 ). 23 ]implementedthesystemofequationsin( 3{1 )inGAPtoshow(G)2forallgroupsGsuchthatjGj1000. 27

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LetCbeaconjugacyclassofaniteperfectgroupGsuchthathCi=G.Intheir1985work,Z.Arad,J.StaviandM.Herzog[ 1 ]demonstratetheexistenceofapositiveintegernsuchthatCn=G.Theminimalpositivesuchintegeriscalledtheconjugacyclasscoveringnumber,anddenotedcn(C).Theconjugacyclasscoveringnumbersofnumerousgroups,includingthealternatinggroups,nitegroupsofLietype,andspeciallineargroupshavereceivedstudy([ 33 ],[ 28 ]).Itisnaturaltoask,givenanyconjugacyclassCofaniteperfectgroupGwherehCi=G,whatupperboundsmaybeplacedoncn(C)?Workhasbeendoneonthisquery,withtherstboundsobtainedinthenotedworkofArad,Stavi,andHerzog.Thereafter,inanapproachusingsemisimplecommutativealgebras,DavidChillag[ 9 ]showsthatifm=n(C) 10 ]obtainsanalternatebound,showingthatwithkthenumberofconjugacyclassesofG,cn(C)k 28

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1 ]. Proof. 4.1.5 )for1kn,wehave, 29

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4{1 )becomes,me11Xi=0kiai=0(0kn): Let2Irr(G).If(C) 4{2 )anda0=1,impliesthatn0,asme11>n.Byequation( 4{2 ),weknowthat, Wehave, 4{3 )mp: 30

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2nnYi=1(jij11)forj=1;2;:::;me11:(4{4) 0=1.Then,theinequalityin( 4{4 )holdsfor1jn.Forj>n, jjijnYi=1j1ij 2nnYi=1j1jijj jij=1 2nnYi=1jij11: 4{4 )istruefor1jme11. Thisimmediatelygivesthefollowingcorollary. 31

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2nnYi=1(jij11)jGj Proof. 4.1.7 ),wemayndapolynomialp(x)2C[x]ofdegreen,suchthatp(1)6=0and, 2nnYi=1(jij11)forj=1;2:::;me1: 4.1.6 )givetheresult. Wenowgiveaninterestingboundforcn(C)ingrouptheoreticterms. 4.1.6 )wehavetheboundcn(C)(me1)e1(C).Wedenoten(g)=jf2Irr(G)j(g)=0gj.Thisgivesjme1(C)jjIrr(G)jn(g)+1.ItisalemmaofGallagher[ 14 ],thatn(g)jIrr(G)jjCG(g)j.Thisgivesustheboundjme1(C)jjCG(g)j+1.Now,wenotethate1(C)o(g)andtheresultfollows. ModifyinganargumentofS.Garrison(Problem4.2in[ 21 ]),wedetermineanupperboundone1(C).Thiswillenableustoplaceanupperboundforcn(C)incharactertheoreticterms. 32

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Proof. 4.1.5 )for1km1,wehave, Settingai=X2Irr(G);(C) 4{5 )becomes, Now,thematrixk1iisaVandermondematrix,sodetk1i6=0,andequation( 4{6 )hasonlythetrivialsolution.However,wenotethatif(C) 4{6 ),acontradiction.So,C1Csforsomeswith1sm1,andhence,e1(C)m. Lemma( 4.1.10 )togetherwithTheorem( 4.1.6 )yieldthefollowingboundoncn(C). 1 ]establishedthatgivenaniteperfectgroupG,inwhichhCi=G,withkthetotalnumberofconjugacyclasses,thenumberofnonidentity 33

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2thenumberofcomplexconjugacyclasses, cn(C)(+)e1(C):(4{7) Then,in1997,inthesamesituationwithm=jm(C)j,DavidChillag[ 10 ]showedthat, cn(C)m22m+2:(4{8) Subsequently,in2005,Chillag,Holzman,andYona[ 10 ]proved, cn(C)(m1)djIrr(G)j TogiveevidencefortheimprovementsthatcanbemadewithourboundswewillconsiderafewsmallnonabelianperfectgroupswherecomputationscanbeecientlyperformedwithGAP.Specically,weconsiderthegroupsA5,SL2(5),SL2(8),theperfectgroupoforder1080,andtheMathieugroup,M11.Foreachgroup,wewilllisttheconjugacyclasseswhichgeneratethegroup,theactualcoveringnumber,andthevariousboundsdiscussedabove.Wedenotetheboundin( 4{7 )asASH,theboundin( 4{8 )asC1,theboundin( 4{9 )asC2,ourboundofcn(C)e1(C)(me11)asB1,ourboundinCorollary( 4.1.11 )asB2,andourgrouptheoreticboundinCorollary( 4.1.9 )asB3.WenotetheupperboundswerecomputedwithaprograminGAPthattakesasinputthegroupinquestion,andreturnsasoutputthegeneratingconjugacyclasseswiththerespectivebounds.Theactualcoveringnumberswerecomputedwithaseparateprogram. Table4-1. BoundComparisonforcn(C)forA5 22810126121032810126121251381716820305238171682030 34

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BoundComparisonforcn(C)forSL2(5) ClassActualASHC1C2B1B2B3 3316172482021421610186122051516504214565552516504214565563162630824421015166548146311010251665481463110 Table4-3. BoundComparisonforcn(C)forSL2(8) ClassActualASHC1C2B1B2B3 22161018612183316172482030712162630103056722162630103056732162630103056913163736124290923163736124290933163736124290 Table4-4. BoundComparisonforcn(C)forPerfectGroupofOrder1080 ClassActualASHC1C2B1B2B3 242250701040503132226501030303232226501030304422265010305651422507014568052422507014568061433145120219115062433145120219115012143382901550156122433829015501561514331451202191240152433145120219124015343314512021912401544331451202191240 Thedatashowsthatforeachconjugacyclassofeachlistedgroup,theboundB1isthestrongest.Ineverycasewiththeexceptionoftheclassofelementsoforder4in 35

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BoundComparisonforcn(C)forM11 231450421248983214373612425742142630103036521417248204063142630824428132137361842728232137361842721113212630153013211232126301530132 9 ]showedtheexistenceoftheintegerifandonlyifZ()=1G.AradandLipman-Gutweter[ 2 ]continuedbyprovidingupperboundsforccn()similartothosefoundforconjugacyclasscoveringnumbersintheworkofArad,Stavi,andHerzog[ 1 ].AradandLipman-Gutweteralsodeterminedthatforany2Irr(G),whereG=Sz(q),q=22n+1,ccn()4.Subsequently,similartohisanalysisofconjugacyclasscoveringnumbers,DavidChillag[ 9 ]showedthatfor2Irr(G)andm()=n(C) 10 ]thatifkisthenumberofirreduciblecomplexcharactersofG,ccn()k 36

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Wenowhaveourmaintheoremregardingcharactercoveringnumbers.ItsstatementandproofaresimilartoTheorem( 4.1.6 ). Proof.

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Settingai=Xg2G;(g) 4{10 )becomes,me11Xi=0kiai=0(0kn): AsZ()=1,a0=(1G) 4{11 ),impliesthatn0,asme11>n.Byequation( 4{11 ),weknowthat, Wehave, 4{12 )mp: 38

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WenowcombineLemma( 4.1.7 )withTheorem( 4.3.3 )toobtainthefollowingcorollary. 2nnYi=1(jij11)jGj Proof. 21 ]),weknowthatthereexists1sm1suchthat[ Lemma( 4.3.5 )inconjunctionwithTheorem( 4.3.3 )immediatelyimplyournalresultoncharactercoveringnumbers. 2thenumberofnonrealirreduciblecharactersby.Similartotheboundinthecontextofconjugacyclasscoveringnumbers, 39

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2 ]establishedthatgivenanitegroupG,2Irr(G)suchthatZ()=1G, ccn()e1()(+):(4{13) Then,in1997,inthesamesituationwithm=jm()j,DavidChillag[ 10 ]showedthat,similartotheconjugacyclasscase, ccn()m22m+2:(4{14) Alsoparallelingtheconjugacyclasscase,in2005,Chillag,Holzman,andYona[ 10 ]proved, ccn()(m1)djIrr(G)j WecontrastourboundsinTheorem( 4.3.3 )andCorollary( 4.3.6 )withthoseabovebyexaminingthecharactercoveringnumbersofthecharactersfromthegroupsA5,SL2(8),PSU3(3)andtheMathieugroupM11.Welabeltheboundin( 4{13 )ALG,theboundin( 4{14 )C1,theboundin( 4{15 )C2,andourboundsinTheorem( 4.3.3 )andCorollary( 4.3.6 ),B1andB2respectively.Again,thecalculationsweredonewithacomputerprogramthatacceptedthegroupandreturnedthevaluesofthebounds.Aseparateprogramcomputedtheactualvaluesofthecharactercoveringnumbers. Table4-6. BoundComparisonforccn()forA5 33817168203381716820428101248528101248 40

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BoundComparisonforccn()forSL2(8) DegreeActualASHC1C2B1B2 741617246157316373610357316373610357316373610358216101848921626301030921626301030921626301030 Table4-8. BoundComparisonforccn()forPSU3(3) DegreeActualASHC1C2B1B2 66183748828751837488287527101803011075271018030110144183748828213182640824213276564247221327656424722731817326152832765642472283276564247232327264015303232726401530 Table4-9. BoundComparisonforccn()forM11 103141724615104283736162810428373616281131426308241632126301530163212630153044214172461545214172461555214101848 41

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42

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WenowgivebackgroundforthechapterconcerningtheTaketaproblemandnormallyseriallymonomialp-groups.WetakeGtobeanitegroupandbeginwithsomecustomarydenitions. ArecordofmanyoftheimportantpropertiesofgroupsofmaximalclassmaybefoundinthegrouptheorytextbyLeedham-GreenandMcKay[ 31 ].Weshallrecallafewoftheirpropertiesthatwillbeofsubsequentuse.Ifweformthelowercentralseries,

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22 ]thatgivenacommutativeringRwithanidentity,thesetofformalpowerseries,

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22 ]calculatedthat, [;]=x+a1b1(mt)xm+t1+:::(5{1) Wenoteasimilarformulaholdsinthequotient,Gn(R). Wenowrecordatheoremthatdescribesacorrespondencebetweencertainp-groupsandLieringsofnilpotenceclassp1.Itwillaidusincollectingasetofexamplesofnormallyseriallymonomialp-groups.ThisisaresultoriginallyduetoindependentworkbyMagnus[ 34 ]andLazard[ 30 ],andwewillrefertoitastheLazardcorrespondence. (i) (ii) (iii) IfHisanormalsubgroupofthegroupG(andthus,LHanideal),theset[H;H]coincideswiththeset[LH;LH]. (iv) ThederivedlengthofthegroupGequalsthederivedlengthoftheringLG. (v) ThenilpotenceclassofthegroupGequalsthenilpotenceclassoftheringLG. 45

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46

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Now,werecordakeytheoremregardingtherestrictionofirreduciblecharacterstonormalsubgroups. Remark5.0.17. jIG()j. WelastlymentionanimportantdivisibilityconditionofItooncharacterdegrees.

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6.2Introduction 40 ]provedthatformonomialgroups,thederivedlengthofthegroup,denoteddl(G),islessthanorequaltothesizeofthecharacterdegreeset,denotedjcd(G)j.Isaacs[ 20 ]andothershaveaddressedthequestion,anditwasconjecturedthatamuchstrongerlogarithmicboundexists.Thatis,thereexistuniversalconstants,C1andC2,suchthattheinequality, dl(G)C1log(jcd(G)j)+C2 24 ],[ 25 ],[ 26 ])eectivelyreducedtheproblemtothecaseofnitep-groups.Keller[ 27 ]thenconsideredthecaseofnormallymonomialp-groups,P,ofmaximalclass.Inthissetting,heprovedtheinequality, dl(P)1 2jcd(P)j+11 2:(6{1) Foranormallymonomialp-group,P,ofmaximalclass,thereisacharacteristicseries, 48

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Denition6.3.1. Wenowgiveaconcreteexampleofanormallymonomialp-groupofmaximalclass. 41 ]showsthatthegroupsGn(Fp),with 49

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29 ]throughadierentmethod.Onemayviewtheirapproachastakingthen-dimensionalvectorspaceL(n)=he1;:::;eni,3npoverFpandimpartingaLiestructureaccordingtothefollowingrelations. [ei;ej]=8><>:(ij)ei+ji+jn0i+j>n: 41 ]showedthatGn+1(Fp)=Mnfor3n8.WeturnLintoaLiealgebrawiththefollowingrelations.[e1;ei]=ei+1for2i7,[e2;e3]=e7,[e3;e4]=e7,[e3;e5]=[e2;e4]=e8,[e2;e5]=e7,[e2;e6]=2e8,and[ei;ej]=0ifnotexplictlylisted.Keller'sexampleistheLazardcorrespondentofthisLiealgebra. Wenowconsiderafamilyofnormallyseriallymonomialp-groupswhicharenotofmaximalclass. 50

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41 ]employedtocalculatethecharacterdegreesetofGn.Weproceedbyinductiononn.Forn=2andn=3,Gnisabelianandtheresultholds.Now,take2Irr(Gn).SupposekerZ(Gn).Then,wemayviewasanirreduciblecharacterofGn=Z(Gn)=Gn1.So,bytheinductionhypothesisisinducedfromKn(b)=Z(Gn),2bn1whenviewedasacharacterthequotient.WeconcludethatisinducedfromKn(b)whenviewedasacharacteroftheoriginalgroup.Now,weconsiderthecasewhen2Irr(Gn)whereker6Z(Gn).SetA=Kn(n 5{1 )thatAisanabeliannormalsubgroup.Consideranirreducibleconstituent,,ofA.Wenote,islinearasAisabelian.Further,YorkdemonstratedtheinertiagroupofisA,andhenceinducesirreduciblytoGn.Then,byFrobeniusreciprocity,06=[;A]A=Gn;Gn.Hence,Gn=andeverycharactermaybeinducedfromalinearcharacterofthegivennormalseries.Now,weaddresstheclassofGn.InanarticleofR.Camina[ 7 ],itisshownthat, [Hi(Fq);Hj(Fq)]=8><>:Hi+j(Fq)ifi6jmodpHi+j+1(Fq)ifimodp:(6{2) ThisleadstoasimilarcalculationinthequotientGn.Wehave, [Kn(i);Kn(j)]=8>>>><>>>>:Kn(i+j)ifi6jmodp;i+jn1Kn(i+j+1)ifimodp;i+jn11Gnifi+jn:(6{3) 51

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Wenowdescribesomeoftheinterestingpropertiesheldbynormallyseriallymonomialp-groups. 52

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jPjj=Pi(1P)=(1P).ThisyieldsPi=andisinducedfromalinearcharacterofPj. jPjjXxPi2 Next,weshowthatanynonlinearcharacterofanysubgroupinthenormalseriesofinductionwill,infact,induceirreduciblytotheentiregroup. 53

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Proof. Proof. 6.3.8 ),P=2Irr(P).So,byFrobeniusreciprocity,1=;PP=[Pi;]Pi:

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jIP()j.Now, jPij(1P)=(1P)=Pi(1P)=tXi=1i(1P):(6{4) Asconjugatecharactershavethesamedegree,( 6{4 )implies,jPj jPij(1P)=jPj jIP()j(1P): Wenowshowthatthecharacterdegreesetoftheentiregroupdeterminesthecharacterdegreesetofeverysubgroupinthenormalseriesofinduction. pa1i; pa2i;:::; pamio; Proof. 6.3.8 ),P2Irr(P).Now,P(1P)=jP:Pij(1P),sojP:Pij(1P)=pajforsomej2f1;:::;mg.Clearly,aj>i,andwehave,(1P)=paj pa1i; pa2i;:::; pamio.Now,let2Irr(P)suchthat(1P)=pajwhereaj>i.Then,asPisnormallyseriallymonomial,thereexists2Irr(Pk)suchthat(1P)=1andP=.AsP(1P)=jP:Pkj(1P)=(1P)=paj,jP:Pkj=paj,andaj=k.Now,asP=PiP=and2Irr(P),Pi2Irr(Pi).Moreover,Pi(1P)=Pi:Paj(1P)=jP:Pajj pa1i; pa2i;:::; pamio. 55

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Proof. 6.3.10 ),cd(Pam)=1; pa1am; pa2am;:::; pamamwhere Wenowgiveatheoremwhichdeterminesthenumberofcharactersofaparticulardegreeintermsofthegrouptheoreticinvariants.ThisgeneralizesaresultofThomasKeller[ 27 ]whichgaveagrouptheoreticdeterminationofthecharacterdegrees(withoutmultiplicity)ofnormallymonomialgroupsofmaximalclass. Proof. ker=\g2P(ker)gP0i:(6{5) 56

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kerP0i1:(6{6) SetA=2Irr(P=P0i)jkerP0i1andB=2Irr(P=P0i)jkerP0i1.Inconsiderationofequation( 6{5 )andequation( 6{6 ),wehavethatA=f2Irr(P=P0i)j(1P)
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6.3.12 )togiveagrouptheoreticdescriptionofjcd(P)j,werecordtworesultsofasimilarspirittoTheorem( 6.3.12 ). Proof. 6{5 )andequation( 6{6 )inTheorem( 6.3.12 )establishthat(1P)=pi.WesetC=f2Irr(P)jkerPk1g.Wenote,asirreduciblecharacterkernelsarenormal,andPhasauniquenormalsubgroupofindexprforr2,wemayviewIrr(P=Pk)=C[D.SimilartotheargumentinTheorem( 6.3.12 ),wehave,jP=Pkj=X2C(1P)2+X2D(1P)2=jP=Pk1j+p2ijDj:

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6.3.13 )).ByLemma2.27in[ 21 ],Z()=ker=Z(P=Pk).Then,jZ()=kerj=pandZ()/P.Finally,asPhasauniquenormalsubgroupofindexprforr2,wehavethatker=PkifandonlyifZ()=Pk1.Thisimpliesthat,jf2Irr(P)jZ()=Pk1gj=pk1(p1) 6.3.13 ). AsacorollaryofTheorem( 6.3.12 ),weobtainthefollowingresult.

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6.3.12 ),jf2Irr(P)j(1P)=pigj>0ifandonlyifP0i1>P0i.Thisgivesthedesiredresult. Wecannowdeterminethesizeofthecharacterdegreesetofanormallyseriallymonomialp-groupingrouptheoreticterms. 6.3.15 )providesalogarithmicboundbetweenthederivedlengthandthesizeofthecharacterdegreeset. 6.3.15 ),P0ak=P0ak+1=:::=P0ak+11>P0ak+1.Byhypothesis,ak+1akb.Now,ama0=(a1a0)+(a2a1)+:::+(amam1):

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35 ]hasproventhatforanormallymonomialp-group,G,withmaximalcharacterdegreepr, dl(G)1+log2(r+1):(6{7) Then,equation( 6{7 )andtheboundonamyield,dl(P)1+log2(bm+2)=1+log2(bjcd(P)jb+2): 6.3.17 )maybeappliedtothefamilyofgroupsmentionedinExample( 6.3.3 ).Fixaprimep,p>2.WeconsidertheLiealgebraL=he1;:::;epi,andnotethatifwedenoteLi=hei;:::;epiandLp+1=0,thelowercentralseriesofLisgivenby,L=L1L3:::LpLp+1=0: 2.Now,wetakePastheLazardcorrespondent,withcharacteristicseries, 2.Then,byTheorem( 6.3.15 ),cd(P)=n1;p;p2;:::;pp1 2o.WemayapplyCorollary( 6.3.17 )withb=1.Thisresultsinthebound, 61

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2+1=log2(p+3): 6.3.17 )inthecaseoftheNottinghamquotientsinExample( 6.3.5 ).Fixp>2andtakeq=prwithr1.AsinExample( 6.3.5 ),weconsidernp+2andGn=Gn(Fq).For1jn,setKn(j)=Hj(Fq)=Hn(Fq)andformthenormalseries,Gn=Kn(1)Kn(2):::Kn(n1)Kn(n)=1Gn: 6.3.5 )thateachirreduciblecharactermaybeinducedfromasubgroupoftheaboveseries.Tomeetourdenitionofanormalseriesofinduction,werenetheaboveseriestoaserieswheresuccessivequotientshaveorderp.Weobtain,Gn=Kn(1):::Pr=Kn(2)::::::P(n2)r=Kn(n1):::Kn(n)=1Gn: 6{3 )inExample( 6.3.5 ),wehave So,fort2f1;:::;ngsuchthat2t+1
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41 ]hasdeterminedthatjcd(Gn)j=n dl(Gn)1+log2rhn SimilartothecaseinExample( 6.3.18 ),equations( 6{8 )yieldthatG(m)n=Kn(2m+11)formsuchthat2m+11n.Hence,thederivedlengthistheminimalintegerm0suchthat2m0+11>n1.Weobtain,dl(Gn)=[log2n]: 6{10 )ismostaccurateforsmallvaluesofr. 63

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[1] Z.Arad,M.Herzog,J.Stavi,PowersandProductsofConjugacyClassesinGroups,LectureNotesinMathematics,Springer.Vol.1112,NewYork,1985. [2] Z.Arad,H.Lipman-Gutweter,Onproductsofcharactersinnitegroups,HoustonJournalofMathematics15(1989)305{326. [3] T.R.Berger,Characterdegreesandderivedlengthingroupsofoddorder,JournalofAlgebra39(1976)199{207. [4] T.Bonner,Productsofcommutatorsandtheorderofanitegroup,JournalofAlgebra320(2008)3165{3171. [5] W.Burnside,Onthearithmeticaltheoremconnectedwithrootsofunityanditsapplicationtogroupcharacteristics,ProceedingsoftheLondonMathematicalSociety1(1903)112{116. [6] W.Burnside,TheoryofGroupsofFiniteOrder,Cambridge,1911. [7] R.Camina,TheNottinghamGroup,Newhorizonsinpro-pgroups,ProgressinMathematics184(2000)205{221. [8] D.Chillag,Characters,nonnegativematrices,andgeneralizedcirculants,ProceedingsofSymposiainPureMathematics47(1987)33{40. [9] D.Chillag,Semisimplecommutativealgebraswithpositivebases,JournalofAlgebra210(1998)33{40. [10] D.Chillag,R.Holzman,I.Yona,Primitivenormalmatricesandcoveringnumbersofnitegroups,LinearAlgebraanditsApplications403(2005)165{177. [11] E.W.Ellers,N.Gordeev,OntheconjecturesofJ.ThompsonandO.Ore,TransactionsoftheAmericanMathematicalSociety350(1998)3657{3671. [12] W.B.Fite,Onmetabeliangroups,TransactionsoftheAmericanMathematicalSociety3. [13] F.G.Frobenius,Uberdieprimfactorendergruppendeterminante,SitzungsberichtederKoniglichPreuischenAkademiederWissenschaftenzuBerlin2(1896)1343{1382. [14] P.X.Gallagher,Groupcharactersandcommutators,MathematischeZeitschrift79(1962)122{126. [15] P.X.Gallagher,Thegenerationofthelowercentralseries,CanadianJournalofMathematics17(1965)405{410. [16] D.Gluck,Boundingthenumberofcharacterdegreesofasolvablegroup,JournaloftheLondonMathematicalSociety(2)31(1985)457{462. 64

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TimBonnerwasborninTrenton,NewJersey.Anortherneruntiltheageofeighteen,heobtainedaBachelorofScienceinchemicalphysicsfromRiceUniversityin2001.AfterashortstayintheDallasbusinessworld,TimtaughthighschoolalgebraforayearatWestsideHighSchoolinHouston,andthenreturnedtotheformalstudyofmathematicsattheUniversityofFloridain2003.HecompletedhisPh.D.intheareaofnitegrouptheoryin2009. 67