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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 pgroups 48 6.2 Introduction . . . . . . . . 48 6.3 Normally Serially Monomial pgroups . . . . .... 49 REFERENCES ...... . . . ............. 64 BIOGRAPHICAL SKETCH ... . . .. ............ 67 LIST OF TABLES Table 41 Bound 42 Bound 43 Bound 44 Bound 45 Bound 46 Bound 47 Bound 48 Bound 49 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 wellknown 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 pgroups, and proceeded to attack this case by a study of normally monomial pgroups 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'blab, 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, (11) 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 (11). 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 ri 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 LipmanGutweter [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 pgroups. Slattery [39] has shown dl(P) < Icd(P) 1 for a pgroup, 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 pgroups, P, of maximal class, and in this setting, he proved dl(P) < 4 cd(P) + 1. We investigate here a more general family of pgroups. 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 pgroups of maximal class are normally serially monomial. Our main result is the following. Theorem E. Suppose P is a noi iniiul ,.ill;l monomial pu.'* 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 pgroup 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 pgroup, 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 pgroups 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 alblab. 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 Gmodule M in the following manner. We take M as the set of ndimensional 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 ndimensional vector space V (turning V into a Gmodule), 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 Gmodule 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 Gmodule 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 Gmodule 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(g1) = 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 ndimensional vector space over C. We may take as a basis the n elements identified naturally with the elements of G. Viewing CG as a Gmodule 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 IG 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 welldefined 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) X2j1 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 (UT1)) 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,...,T7l EG X(u) 2n(1) The last step follows by induction. We complete the proof of (31) by proving a more general theorem. We count the number of viv 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) G21 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 (31). 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) = G2n1 =1 x 2n1 xE Irr(G) xEIrr(G) 1Gn(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). (32) xeIrr(G) X(1) Setting  G  ai= X X (g)( xeIrr(G), x(l)= i we rewrite the system of equations in (32) 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. (33) 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 (33), implies that n < r and mi > CG'. Set .1,. = max{p(Ai) ,..., p(Ar) }. Note .1,. > 0, as r > n. By (33), we know that, ap(Ai) = p(l). (34) i= 1 We have, xEIrr(G), x(l)>l i1 > G ap(Ai) Sl1 Il il^) SIG ) by (34) 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. (35) Pini) +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 (f21)+(1)n+1 For 1 < j < n, p(Aj) = c, and p(l) = 1, so the inequality in (35) 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 (36) 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, (36) 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 (36) 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 i2 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 i1 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 pgroup, 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, n1 G' > 2 (f i= 1 and the result follows. n t) t > 2 (2 _t 1= (n + 1)! (n n+2 n+t (37) 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 pgroup, 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, nl nl pa G' > f? > p2i (n1) 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 CG 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 IG G G and we have the limit in the second statement. Finally, suppose G > 1000 and (G) > 20. Then A(G) > 4. By (38), < (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, LC. Kappe and R. Morse [23] implemented the system of equations in (31) 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. (41) xrr( (1)ke(C)1 XEIrr(G) Setting xEIrr(G), ( y) (c)=a the system of equations in (41) 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, 01. Y aiq(ai) = 0. (42) 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 (42) 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 (42), we know that, Tme 1 Saip(a ) p(l). i 1 XEIrr(G), x(1G)>l me l *, aip(ai) i= 1 > P by (43) > mp. Alternatively, We have, (43) 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 (44) holds for < Then, the inequality in (44) 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 2ail 2TH a 2n >^^>^ ^^ii So, the inequality in (44) is true for 1 < j < m,,  This immediately gives the following corollary. (44) 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, amei}, where ao = 1 and Ia+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 m1 where ao = 1. Then, ei(C) < m. Proof. Let C1 be the inverse class of C. We claim that C1 C C for some s with 1 < s < m 1, and the result will follow. Suppose the claim is false. Then, C1 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 < (45) XEIrr(G) Setting ai= ()X(CcG), EIrr(G), X(C) X(OG) the system of equations in (45) becomes, m1 lai = 0 (0 < k < m 1). (46) i=0 Now, the matrix (a 1) is a Vandermonde matrix, so det (ak1) / 0, and equation (46) 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 (46), a contradiction. So, C1 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). (47) Then, in 1997, in the same situation with m= m(C), David Chillag [10] showed that, cn(C) < m2 2m + 2. (48) Subsequently, in 2005, Chillag, Holzman, and Yona [10] proved, (Irr(G)t cn(C)<(m 1)(r 1 +). (49) 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 (47) as ASH, the bound in (48) as Cl, the bound in (49) 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 41. 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 42. 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 43. 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 44. 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 45. 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 LipmanGutweter [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 LipmanGutweter 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 yEG Setting S09(g1) the system of equations in (410) 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. (411) 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 (411), implies that n < m,, 1 and mrn > G1. Set .1,. = max {p(ai) ,..., p(ame1) Note .l,. > 0, as me, 1 > n. By equation (411), we know that, me11 Saip(ai) = p(l) (412) i= 1 We have, O(1 ) 1 me11 0(91) 5n,1 a I r aip(ai) gEG,gS1c 1 1 > r4PM by (412) > 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 LipmanGutweter [2] established that given a finite group G, X c Irr(G) such that Z(x) = 1G, ccn(X) < e (X) (A + p). (413) 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. (414) Also paralleling the conjugacy class case, in 2005, Chillag, Holzman, and Yona [10] proved, ccn() < (m 1) (rr) + ) (415) 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 (413) ALG, the bound in (414) Cl, the bound in (415) 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 46. 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 47. 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 48. 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 49. 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 ahbi; 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 pgroups. 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 wellstudied family of pgroups 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 LeedhamGreen 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 > > Pn1 > 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+t1 + ... (51) We note a similar formula holds in the quotient, G,(R). We now record a theorem that describes a correspondence between certain pgroups and Lie rings of nilpotence class < p 1. It will aid us in collecting a set of examples of normally serially monomial pgroups. 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 pyin ';,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(ghg1) 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 Gconjugates of 0, the orbitstabilizer 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 pgroups 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 pgroups. Keller [27] then considered the case of normally monomial pgroups, P, of maximal class. In this setting, he proved the inequality, 1 11 dl(P) < cd(P) + . (61) 2 2 For a normally monomial pgroup, 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 pgroups 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 pgroups. We also examine interesting character theoretic properties satisfied by normally serially monomial pgroups. 6.3 Normally Serially Monomial pgroups Definition 6.3.1. We define a pgroup, 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 pgroup 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 pgroup 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 pgroup 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 pgroup 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 pgroups. It also establishes that the derived length of normally serially monomial pgroups 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. LeedhamGreen [29] through a different method. One may view their approach as taking the ndimensional 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 pgroup 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 pgroups 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 (51) 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 (63) 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 pn2 r(f2). So, G, is not of maximal class. We now describe some of the interesting properties held by normally serially monomial pgroups. 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 Pconjugate 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 pji'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(xgx1) 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 pi/;,'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 Pconjugates of x and t = IPI. Now, IPW XP(1p) X(lp) = (lp) = Op(lp) Yx(1p). (64) 1 I i= 1 As conjugate characters have the same degree, (64) 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, pai.,.. pai , where for j e {1,..., m}, pa}i = paji if aj i > 0, and paji = 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... pai 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 ri pa2i. pai. This establishes the following corollary regarding abelian subgroups of maximal order. Corollary 6.3.11. Suppose P is a no malbil ,.ill.; monomial pj,.;,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, pala, pa2a,..., pama, } where pja  pajam if aj am > 0, and pajm = 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> .. > P1 > 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,'. (65) Now, suppose ker6 > P1. 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 = pi1. This is a contradiction, so, ker 0 P1__ (66) Set A {e Irr(P/P)  kery > P1_ consideration of equation (65) and equation (66), 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 (65) and equation (66) in Theorem (6.3.12) establish that 0(lp) = p'. We set C = {0 e Irr(P)  ker 0 > Pk1k}. 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 Pk1 +2"i  . XEC XED Solving for D, we get, D P : PkP P: P1 p2i P/Pk1 (P ) P2i p1 (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) Pk1} {X c Irr(P)  ker = Pk}. In particular, pk1 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 pgroup 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) = Pk1} (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,...,n1} 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 pgroup in group theoretic terms. Corollary 6.3.16. Suppose P is a nc mll;l .I ,.!ll;i monomial pip.;'r, with normal series of induction, P = PO > P > ..I > PI > 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 pi'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 Pak11 > 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 pgroup, G, with maximal character degree p', dl(G) < 1 + log2 (r + 1) . (67) Then, equation (67) 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 