Fusion of p-elements in symmetric groups

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Fusion of p-elements in symmetric groups
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Thesis--University of Florida.
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Includes bibliographical references (leaves 68-69).
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by John-Tien Hsieh.
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FUSION OF' -ELEMENTS IN
SYN v1M'R. C GROUPS






BY

JOHN-TIEN HK;EH


A DISSERTATION PRES[:iTLE) TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1978















TO MY FATHER


TENG-CHUAN HSIEH










ACKNOWLEDGEMENTS


I wish to express my gratitude to Professor Brooks,

Professor Bullock, Professor Drake, Professor Farmer and

Professor Hale, who are members of my supervisory committee,

for the constant help and encouragement they gave me during

the years of my study in the University of Florida.

I am specially grateful to Professor Hale, who is

my advisor, for his continuous guidance of my work toward

the goal of my research. Without his excellent instruction,

this work would not be done.

I wish also to thank the Mathematics Department of

the University of Florida which has given me the financial

support for my studies and to thank all faculty members,

staff and graduate colleagues of this department for the

great experience I received from them during the years I

have studied in this university.


iii












TABLE OF CONTENTS



ACKNOWLEDGEMENTS........................................ iii

SYMBOLS........................................................ v

ABSTRACT ................................................vi

CHAPTER

I INTRODUCTION.......................................1

1. Simple Groups and Finite Groups............ 1
2. Local and Global Properties of Finite
Groups.................................... 2

II SYLOW SUBGROUPS OF SYMMETRIC GROUPS ................8

3. Symmetric Groups........................... 8
4. Direct Products, Semidirect Products
and Wreath Products .....................10
5. Sylow Subgroups of Symmetric Groups....... 13
6. Structure of S n ..........................16

7. Normalizers and Centralizers..............26

III FUSION OF ELEMENTS IN Sn I........................33

8. Cycle Structures and Conjugacy Classes....33
9. The Main Theorem........................... 34

IV FUSION OF ELEMENTS IN S II.......................45
pn
10. The Second Theorem...... .................45
11. Fusion in Direct Products .................55

APPENDIX...................................................... 64

REFERENCES............................................... 68

BIOGRAPHICAL SKETCH......................................70










SYMBOLS


The following expressions concerning a finite group G will
be used frequently:

E identity element of G
IAI the order of a set A
ACG A is a subset of G
A AgG A is a normal subgroup of G
Z(G) center of G

For H N(H) normalizer of H in G
C(H) centralizer of H in G
[G:H] index of H in G

For H,K
NH (K) N(K)nH

CH(K) C(K)nH

For NIG, we define:

G/N the factor group of G over N

We also define; for A,BcG, H,K,L
A-B set of all elements in A not in B
subgroup of G generated by A
h9 g-lhg
Ag g-1Ag
[g,h] g -h-gh
[H,K] < [hi,k] h.H, k,-K
G [G,G]











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



FUSION OF p-ELEMENTS IN
SYMMETRIC GROUPS

BY

JOHN-TIEN HSIEH



DECEMBER 1978



Chairman: David A. Drake
Cochairman: Mark P. Hale Jr.
Major Department: Mathematics



In a finite group G whose order is divisible by a prime

p, the normalizers of p-subgroups are called local subgroups

of G. The significance of local subgroups in the discussion

concerning the fusion of p-elements, hence the relations

between local and global properties of G, has been recognized

in recent years.

In this paper the author defines a local fusion family

of a Sylow p-subgroup S of G to be a family containing nor-

malizers of some p-subgroups of S such that if any two ele-

ments of S are conjugate in G then the conjugating element

can be found as a product of elements from local subgroups

of the family.







It is proved, as the main theorem of this paper, that

in the Sylow p-subgroups of a symmetric group of p-power

degree, the normalizers of p-subgroups of only two differ-

ent structures are needed to form a local fusion family.

In fact, they are the normalizers of a large elementary

abelian group and some small elementary abelian groups of

order p

Besides the main theorem which is proved in two

parts, the structures of the normalizers of the two

p-subgroups mentioned above are described in full detail.


vii












CHAPTER I
INTRODUCTION



1. Simple Groups and Finite Groups

In the study of finite groups much of the work centers

on the study of finite simple groups, that is partially

because simple groups to finite groups are just like prime

numbers to natural numbers. Knowing all about simple groups

would tell much about all finite groups. The Jordan-Hilder

theorem for finite groups says: For every group G there is

a sequence of subgroups G=Go>G1>... Gr_>G r=E, such that

each quotient group Gi/Gi+1 is a simple group and the collec-

tion of associated simple quotient groups is unique up to

reordering. This is somewhat analogous to the Fundamental

Theorem of Arithmetic: For every positive integer n/l there

is a sequence n=n o>nln2>-*. nr-_lnr=1 such that each

ni/ni+1 is a prime and the collection of primes which so

occur and their multiplicities are uniquely determined up

to possible renumbering of indices i.

Suppose all finite simple groups were known. How could

one use this and the Jordan-Holder Theorem to determine all

finite groups? First one would get a list of simple groups

Sl,S2,...,Sr, then would try to build up a tree diagram

listing the possible groups Gi such that Gi/Gi+l Si+l,

ending with all possible G=Go whose composition factors are







the simple groups Sl,S2,...,Sr. To proceed doing this, in each

step one would have to know how to determine all possible

G. by knowing G and Gi/G +S i+l. This question is the

so called Extension Problem which can be stated in a more

general way: Given two groups N and H, determine all possible

groups G such that NiG and G/N=H. Such groups G are called

extensions of N by the factor H. All possible such extensions

G can be constructed in some systematic way. Schreider [19]

developed a technique using automorphisms of N and a factor

set to construct an extension G by obtaining the multipli-

cation table for G. This method can also be translated into

the language of cohomological algebra in the case that N

is abelian. (See Rotman [18] 5 and 10 or Huppert [13]

Ch I, 16-17.) Thus Schreider s method and the Jordan-Holder

Theorem allow us to construct all possible finite groups,

although not very practically, from all finite simple

groups. This is why simple groups are so important in

finite group theory.



2. Local and Global Properties of Finite Groups

For every subgroup H of a finite group G, the

structure of H, such as the list of elements and subgroups

and the equations of the multiplication table of H, or the

embedding of H in G are part of the local structure of G.

While the normal subgroups, quotient groups and conjugacy

classes are relevant to the global properties of G. The re-

lations between the local and global properties of a

finite group are very important. In the study of these





relations the representation theory and transfer are very

useful. The application of these techniques is often depen-

dent on results concerning the fusion of elements. The study

of fusion of p-elements, where p is a prime, is also very

close related to the question of whether a given group G

possesses a nontrivial p-factor group, which is in turn

related to the question of simplicity of finite groups.

Higman (1953), using the transfer, proved the focal

subgroup theorem; see [12]:

(2.1) If P is a Sylow p-subgroup of a finite group G,

then PrG' is generated by all elements of the form a-lb

where a and b are elements of P conjugate in G.

This theorem shows that the fusion of elements of P

determines the focal subgroup PAG', hence P/PAG', which

is isomorphic to the largest abelian p-factor group of G.

Thus the fusion of elements of P plays an important role

in connections of local and global properties of G.

The question of how two elements of P are fused in G

has a very satisfactory answer stated in Alperin's fusion

theorem [1]:

(2.2) If P is a Sylow p-subgroup of G and a,b are elements

of P conjugate in G then there exist elements a=aj,a2,...,

am=b and subgroups H ,H2,...,Hm_1 of P such that ai,ai.1

lie in Hi and are conjugate in the normalizer N(Hi) of H.,

i=l,2,...,m-l.

The well-known theorems of Burnside, Frobenius and

Grun giving conditions for the existence of nontrival

p-factor groups can be proved much easier by the focal

subgroup theorem and Alperin's fusion theorem. For more







detail about the proofs of these theorems, see Gorenstein

[9] Ch 7.

If we call the normalizer of a p-subgroup a local

subgroup, then Alperin theorem can be interpreted as: If

two elements of a group are conjugate, then the conjugating

element factors into a finite number of elements, each lying

in a local subgroup. Each such element acts as a local

conjugation.

The significance of local subgroups in the fusion of

p-elements can be seen further in Alperin's recent article

[3]. Other questions such as the number of local conjugation

steps and the family of p-subgroups whose normalizers give

all local fusion were discussed extensively by some people.

As to the first question, Alperin's conjecture that the

number of local conjugation steps is bounded by a function

which depends only on the nilpotency class of P is proved

by Dolan in [5]. Alperin in [2] (1974) also proved that in

his theorem, the elements al,a2,...,am can be chosen so that

the orders of their centralizers in P first increase monoto-

nically and then decrease monotonically; this is called "up

and down fusion". As to the second question, Alperin in [11

also defined a conjugation family and weak conjugation

family as follows:

(2.3) Let P be a Sylow p-subgroup of G, a conjugation

family F (resp. weak conjugation family) for P is a col-

lection of pairs (H,T), where HP and T5N(H), satisfying

the following condition: For any two subsets A, B of P with

B=Ag for some g in G, there exist (HI,T ),(H2,T2),...







(Hn,Tn) in F and elements xl,x2,...,x ,y in G such that

(a) g=xlx2*-x..n (resp. AXX2 ..XnY=B)

(b) xi:Ti, l
(c) A 1' -i+l
The general form of Alperin theorem is that the family

F of all pairs (H,T), where H is any Sylow intersection of

the form H=PnQ with Q also a Sylow p-subgroup of G and both

Np(H) and N (H) are Sylow p-subgroups of N(H), T=N(H), is a

conjugation family for P. These Sylow intersections H are

called tame Sylow intersections. Some examples of conjugation

families and weak conjugation families are also given in [1];

Among them the family FC={(H,T)} where H=PnQ is a tame

intersection and T=C(H) if Cp(H)VH while T=N(H) if Cp(H)
is a conjugation family, this means that the fusion pattern

of such subgroups H in N(H) with C (H) p -
complete fusion pattern of subsets of P in G. Goldschmidt

in [8] showed that the fusion is in fact determined by the

p-local subgroups N(H) where N(H)/H is p-isolated.

For fusion of subsets in a p-subgroup of G, rather than

a Sylow p-subgroup of G, one can find some account in Kantor

and Seitz's article [16]. Also more theorems about local and

global properties of finite groups can be found in the foll-

owing references:Alperin and Gorenstein [4], Finkel [6],Hall [10],

Glauberman [7] and Thompson [20].

Intuitively, a good conjugation family requires either

fewer members in the family or more groups in the family are

of known types.









A family F of local subgroups is said to control the

local fusion of the Sylow p-subgroup P if :

(2.4) Whenever two elements x, y of P are conjugate in

G, then there exist N,N,,...,Nr in F and elements g,g,,

...,g such that g.EN i=l,2,...,r and

xgl2" r=y.

We will call such a family a local fusion family, note

that here in the local conjugation steps, each pair of

elements is not required to lie in the p-subgroup whose

normalizer in one of N..
1
The area of interest of this paper is to examine the

fusion of elements in the Sylow p-subgroups of the symmetric

group of any degree and to find some "good" local fusion

families of the Sylow p-subgroups. That is, to find some

families of p-subgroups of G whose normalizers control the

fusion of p-elements in G. As was proved by Alperin, the

normalizers of all p-subgroups of the Sylow p-subgroup P

is certainly one such family, but it is much too big to be

significant. The significance of this paper is its discovery

of some very nice local fusion families for the Sylow

p-subgroups of the symmetric group of a p-power degree.

Each such family contains normalizers of p-subgroups of

only two different structures. In fact, they are one

"large" elementary abelian group and many "small" ele-

mentary abelian groups of order p2

In order to deal with permutations of high degrees and

look into their cycle structures we have to use many new








notations which work as language of the discussion and make

the theory of this paper sound.

Only knowledge of fundamental group theory such as

Sylow theorems is needed. Some counting methods are also

used in the proofs of the theorems. In chapter II Sylow

subgroups of symmetric groups are discussed, some elemen-

tary results are given, and a complete discussion of Spn

(the Sylow p-subgroup of the symmetric group of degree

pn) as wreath products of Spn-l and C (cyclic group of

order p) in two ways are shown. Also some group-theoretic

terminologies are introduced. The main theorem is proved in

two parts; the first part is proved in Chapter III and the

second part is proved in Chapter IV. Some examples illus-

trating the proofs of these theorems are given in the

appendix. All groups are considered finite in this paper.











CHAPTER II
SYLOW SUBGROUPS OF SYMMETRIC GROUPS



3. Symmetric Groups

Given any set n of order m, a one-to-one map from 0

onto itself is called a permutation on p. The set of all

permutations on p forms a group under the composition of

maps, called the symmetric group of degree m, and denoted

by m It is a simple combinatorial matter to see that

ml =m!. Any subgroup of ym is called a permutation group
of degree m. Cayley's theorem (see [11] 2.9) shows that

every finite group can be realized as a permutation group,

that is, every finite group has a representation as a permu-

tation group. This clearly shows that the symmetric groups

themselves merit close examination.

Let Y be a symmetric group on the set ? of m elements.

For each xeym, x induces a decomposition of Q into disjoint

subsets, namely, the equivalence classes of the equivalence

relation: xg6 if and only if there is an integer i such that

B=ax These disjoint subsets are called orbits under x.

If aEn then the orbit of a under x consists of all elements

axi with i=0,+1,+2,..... For each a.E there is a smallest

positive integer d dependent upon a such that ax =a ; thus

the orbit of a under x consists of exactly d elements: a,ax,

ax2,...,axd-1. By the cycle of x containing a we mean the











2 3 d-l
ordered set (a,ax,ax ,ax,...,ax d-) considered as the

restriction of the map x on this orbit. The integer d is

called the length of the cycle. The commas "," between

elements in the cycle are usually omitted. For example

if a=l, ax=5, ax2=7, ax =3, ax4=l, then the cycle of x

containing a will be (1573). Two different cycles on the

same orbit may represent the same permutation; for example,

(1573), (5731), (7315), (3157) all represent the same map

on the orbit {1,5,7,3}. For each permutation x and alE

the symbol x:a-- means x maps a to 3. We will also use
x
a^ = to mean the same thing. The following proposition con-

cerning the permutations are elementary and its proof can

be found in most algebra or group theory texts:

PROPOSITION 2.1. Let y be a symmetric group on o

and x,y,z,Eym then

(1) The order of x is the least common multiple of the

lengths of disjoint cycles under x.

(2) If z-lxz=y,(i.e. xZ=y),then x and y have the same

cycle structure; that is, there is a one-to-one correspon-

dence between orbits of x and orbits of y which preserves

the length of orbits.

(3) If x,y are conjugate in G=Ym, (i.e. there is zEG such

that xZ=y) and if

(3.1) x=(ala2...a )(blb2. .b n2)...(lx2... x nr

Y=(ala2...anl) (1621 ..* n2) ...(X X2 Xnr)






are cycle decompositions of x and y (as products of disjoint

cycles), then the element of g sending ai to ai, bj to j

..., xe to e for all i, j, ...e is a conjugating element

for x to y, (i.e. xg = y).



4. Direct Products, Semidirect Products and Wreath Products

The terminologies contained in the following definition

are needed:

DEFINITION 2.2. Let A, B be two groups

(a) If A, B are subgroups of a group G with G=AB and AnB=E,

we call the subgroup A a complement to the subgroup B in

G and A,B are said to be complementary in G.

(b) By the direct product of A and B we mean a group G

containing two normal complementary subgroups A*, B* with

A*=A and B*=B. G is denoted by G=AxB.

(c) By a semidirect product of A by B (or a split extension

over A by B) we mean a group G containing a normal subgroup

A* isomorphic to A and a subgroup B* isomorphic to B, such

that A* and B* are complementary in G. We write G=[A]B

to mean G is a semidirect product of A by B.

(d) Let A be a group and B a permutation group on Q with

lol=n, then the wreath product AiB of A by B is the set

(4.1) G={(f,b)/f:Q-+A, bEB}

with the multiplication

(4.2) (flbl) (f2,b2) = (g, blb2) where g=Q-A is defined

g(0)=fl(")f2 bl) for all DEn. (Here Dbl represents the

image of a under the permutation bl, while we use g(D) to

to denote the image of a under the map g).




11

It can be shown the multiplication is associative

and the element (f,E), where f(a)=E all aen, is the identity

of the wreath product and the inverse of the element (f,b)

(4.3) (f,b)-l=(g,b-l) where g(a)=[f(b-l) ]- aE.

Hence G is a group. The group B is called the "top"

group, A is called the "bottom" group of the wreath product.

PROPOSITION 2.3. Let B be a permutation group on the

set q={1,2,3.,...,n} and A a group. Let G=A1B be the wreath

product of A by B, then

(1) G contains a normal subgroup N=AxA2x...xAn

(4.4) Where Ai={(f,E) If(a)=E for all 3ai} =A

(2) The subgroup B*={(e,b) beB, e(a)=E all DeQ} is iso-

morphic to B and G=NB* with NnB*=E, that is, G is the semi-

direct product of N by B, i.e.

(4.5) G=[AxAx...xA]B and hence

(4.6) IAIBI=IAInIBI

(3) If A is also a permutation group on the set r,then

G=AIB is a permutation group on rxo with the action:

(4.7) (y,) (f,b)=(yf()),Db) for each yEF and aDE

and (f,b)EG. (Here, Db means ab, yf(a) means y )

(4) For any permutation groups A on F, B onQ and C on A

(AIB)IC and Ai(BIC) are permutation groups on (FxP)xA

and Fx(QxA) respectively and

(4.8) (AIB) IC=A (BIC)

Thus, the wreath product "I" as composition between

permutation groups, is associative. We will prove only

part (3), the proofs of other parts can be found in Huppert

[13] Ch. 1 15.




12

PROOF: (3) For each (f,b)EG, (y,)cErxQ define

(f,b)* on rx2 by

(4.9) (y,3)(f,b)*=(yf(9), ab) then

(i) (f,b)* so defined is a one-to-one map from FxQ

into FxQ: For let

(yl,9l) (f,b)*=(Y2, 2) (f,b)*

then (Y1f(al),I) b)=(y2f(2) ,2b)

so Ylf(11)=Y2f(22) and ib=2b

Since b, considered as a permutation on 9, is one-to-one,

D1=32. This implies ylf(01)=Y2f( 8), again because f(Dl)
is a permutation onF and hence one-to-one. So we have

Y1=Y2. Thus (Yl,'l)=(Y2 ,2) and (f,b)* is a one-to-one

map.

(ii) (f,b)* is onto: For each (3,y)ErFx.

(4.10) Let 8*=ab-1 and y*=y(f(,*)-1 then

(y*,D*)(f,b)* = (y*f(D*), 8*b)
-1 -1
= (Y. (f(D*)) f(9*), Bb b)

= (y,2)

So (f,b)* is onto map.

(iii) The map (f,b) -(f,b)* is a homorphism: for let

(fl,bl) (f2,b2)E G=AIB. We need to show

(4.11) (fl,bl)*(f2,b2)* = [(fl,bl)(f2,b2)

For each (y,9)EFxQ

(YV,) (fl,bl)*(f2,b2)* = (yfl() ,bl) (f2,b2)

= [(Yfl(9)]f2(9bl), blb2)

= [Yg() D(blb2)]

Here g(D)=f ()f2( bl) works as the function g defined in
H zre g~) =defined ina






(fl,bl)(f2,b2)=(g,blb2) in (4.2)

thus (y,3) (fl,bl)*(f2,b2)* = (y,a) (g,blb2)*

= (y, ) ( (f ,bl) (f2,b2) )*

So (fl,bl)*(f2,b2)* = ( (fl,bl) (f2b2))*

and (f,b) + (f,b)* is a homorphism.



5. Sylow Subgroups of Symmetric Groups

Given a positive integer m and a prime p, there exist

non-negative integers aO, al...,ar, all less than p, such

that

(5.1) m=arpr+ar-p r-l+...+a3p 3+a2p2+ap+ao

It is well known [17] that the number of powers of

p which appear in the product m!= m(m-l)(m-2)...3.2.1 is

(5.2) e(m)=ar(l+p+p2+...+pr-l)+ar-i(l+p+p2+...+pr-2)+...

+a3(l+p+p2)+a2(l+p)+al

This means Imlp=pe(m), if we uselm|pto denote the

maximal.p-power dividing m, for any integer m and prime p.

Hence, the order of the Sylow p-subgroups of the symmetric

group Ym on a set 0 of m elements is pe(m)

Let C be the cyclic group of order p and let

(5.3) InC=CiCiC...iCn times for all n=1,2,3,... let

OC=E. The proof of the following proposition can be found

in [141 by Kaloujnine 1948:

PROPOSITION 2.4. If the cyclic group C of order p is

considered as a regular permutation group on the set

o= {1,2,3,...p}, then

(1) nC is a Sylow p-subgroup of the symmetric group Ypn

of degree pn on the set Qn=QXXx...xQ, n times.







(2) The Sylow p-subgroups of the symmetric group Ym where

m satisfies (5.1), are isomorphic to the

(5.4) group (IrCxlrCx...xl C) x (ir-lCxlr-1Cx...xlr-1C)

ar times aritimes

...x (12Cxl Cx...x 2 C) x (11Cxi Cx.. .xiC) x (1 CxioCx...I C)

a2 times al times a times

or, in short,
r i i r ai i
(5.5) nT (liCxi Cx...xl C) or n ( in C) .)
i=o ------ i= j=1
ai times

This proposition shows that the sylow p-subgroups of

symmetric groups of any degree are direct products of sylow

p-subgroups of symmetric groups of p-power degrees. This

fact motivates the close study of the sylow p-subgroups

InC of the symmetric group Y n for any positive integer n.

By the Sylow Theorems, all sylow p-subgroups of a

finite group are conjugate; finding the structure and the

fusion pattern of one sylow p-subgroup will tell the struc-

ture and fusion pattern of all sylow-p-subgroups. From

now on, we will be studying a particular sylow p-subgroup

of Y n on the set n of pn elements.
p n
Let Qn be the vector space of dimension n over the

(5.6) field Q={1,2,3,...,p}

which is isomorphic to the field {Z ;+,.} of integers

modulo p. (We will use both the symbols p and 0 to denote

the class [0] in {Z ;+,.} We consider Qn as Qxlx...xn
p n
n times. i.e.

(5.7) n={ [V1v'l2, ...V] viE i=l,2,...,n} with

component-wise addition and multiplication.




15
For each j=l,2,3,...,n-l, 0j will denote the j-

dimensional subspace QxQx...xnxx0x0...x0

(5.8) ={ [vl'v 2, ...v.j,0,0,' ...' 0] I viEQ,i=1,2,...,j}

An arbitrary vector in n will usually be denoted

by v(n)=[vl,V2,...vn], or vn. Here we use the bracket ]

to denote vectors in order to be distinguishable from the

( ) which represents cycles. If j is the largest i such

that v. A 0,then v(n) is labelled by v j (or vJ) and is

considered as a vector in the j-dimensional subspace Qj.

The superscript (j), which represents the dimension, will

be omitted if it is clear what dimension is being discussed.

Let y n be the symmetric group on the set Qn and let

S n be the set of all permutations which are defined by

(5.9) x:[vl,v2...,vn] 1 [v +o, v2+xl(v1), v3+x2(vl1v2),

v4+x3(vl,v2,v3),...Vn+xn-l(vl,V2,'...,vn-)] for all

v=vn=[vl,v2,...,vn]Ein

where XoC0, x1= Q, x2=Q2 0,...

xn-l;n_-1 -' are arbitrary maps. x so defined is denoted

by

(5.10) x={x ,x' x2, ... ,x n-

Kaloujnine in [15] proved that Spn defined above is

a group isomorphic to a sylow p-subgroup of the symmetric

group pn. Our permutations are written on the right

i.e. the image of a vector v under the permutation x is

written vx. While the functions xi in (5.9) are written

on the left, e.g. x (v ,v2-...vi) represents the image of

the vector [vlv2-*...i ,v]E under the map xi.






6. Structure of S n
p
Let us define and examine the generating elements of

S n and some other elements and subgroups which play im-

portant roles in this paper:

For each j
(6.1) tv(j)={O ,O ,...,O 0 (j), 0,0,...,0}

j n-j-1

where v(j):nj -4 is defined by

) 1 w() = v(
(w ) otherwise
t(j moves only vectors with v(j) as their first j entries

and maps [v ), vj *,*,...,*] to [v(j), v* ..,*

here denotes arbitrary entries which are not related to

what is being discussed. Let

(6.2) t = {1,0,0,...,0} which maps [v1- ... v ] to

[v+1,v2, ... ,v for all [v1... ,vn] e. For each j
v(j):j, let

(6.3) Tv(j) = the subgroup generated by {t[v(j),a] aen

and

(6.4) t (j)* = t t which is the generator of
v aeQ [v(J) ,a]

the diagonal subgroup of the group T (j). Finally,

(6.5) t* = t = {0,1,0,0,...,0} ,
0 aEQ [a]
For each jin-l, let

(6.6) Aj = v( T V(j-) = the subgroup generated


by all tv(j) with v(jQ.. In particular, we use T to denote

An-1 which is the subgroup generated by all t (n-1)'s, v(n-l)
v
en-1. i.e.






(6.7) T = An-1 = (n-)n


It is the subgroup T which plays the key role in the

main result of this paper. Let us describe the above nota-

tions graphically:

Consider a "sun" sitting in the center of a "big"

circular orbit of p primary planets, each planet represents

a number in Q. Then consider each primary planet as

sitting at the center of a secondary orbit containing p
2
secondary planets each represents a vector v in n2.

For example, the planet corresponding to v2=[2,3] is the

number 3 secondary planet sitting on the secondary orbit

which around the primary planet numbered 2. See Figure 2.1:














Z'3 1-~2, 32

Figure 2.1

Then for each secondary planet, there is an orbit

(called third order orbit) centered at it which contains

p third order planets. Each represents a vector v E 3'

Proceed this until the "nth order" planets sitting on the

"nth order" orbits around the (n-l)th order planets. See

Figure 2.2.






4, OAb PCve^t


Nb {\J;J (p1) eefe pl&di3







Figure 2.2

Thus, we have a "Galaxy" which contains a complex

chain of orbits and planets. The element to represents

a transformation of the Galaxy obtained by a 1/p-revolution

of the primary orbit. (In the same direction as the order

the planets on the orbit are labelled.)

For each aen, t[a] represents the transformation

obtained by a 1/p-revolution of the secondary orbit around

the primary planet corresponding to [a]. In general,

for each j
formation obtained by a 1/p-revolution of the (j+l)th

order orbit around the jth order planet corresponding to

the vector v(j). t* represents the transformation ob-

tained by rotating all p secondary orbits 1/p-revolution
(j) *
simultaneously. In general, for each v e. t

represents the transformation obtained by rotating 1/p-

revolution simultaneously all p (j+2)th order orbits

which are around the p (j+l)th order planets sitting on




19

the (j+l)th orbit centered at the jth order planet which

corresponds to the vector vj. The subgroup of T is the

group of transformations obtained by any combination of

rotations of all pPn-1 nth order orbits, each a multiple

of 1/p-revolution. We have

PROPOSITION 2.5. (1) T is an elementary abelian

group of order ppn-

(2) All t's and t*'s are of order p.

(3) A(j) is an elementary abelian group of order pP .

(4) Each T (j) is an elementary group of order pP.

PROOF: Trivial.

PROPOSITION 2.6. (1) If i.j and vini, wJ3e,
1 jL
then t and twi commute if and only if i=j or vi doesn't

coincide with the first i entries of wj.

(2) If i
the first i entries of w i.e.

(6.8) If w= [v Wi+1, wi+2,...,wj], then

(6.9) (t j)vi = tv where v = wJ+l1 e

(6.10) with 1i = [0,0,...0,1,0,0,...,0]
1+1
(i+l)th entry

PROOF (1) We first prove the "if" part-

(a) Assume i=j and vi wj. Since t i fixes all vectors

except those containing v as first i entries and twj fixes

all vectors except those containing wJ as the first

j (=i) entries, every vector is either fixed by tvi

or tw Thus, tvi twj = twj tvi.
wJ v wjl wj Vi
(b) Assume i
i entries of w Then with the same reason as in (a),






every vector is either fixed by vi or by w ; hence t i and

twj commute, The "only if" part of (1) will be shown

after we prove part (2):

(2) Assume i
wi as in (6.8). Since twj: [vi, wi+ ...,wj,wj+l,*,*,...,*]>

[v ,wi+l, ... w,wj,w+1,*,*,...,*] (vectors not mentioned

are fixed). By part (3) of proposition 2.1,

(twj)tvi: [vi,wi+l,...,wj,w j+l *,*,...,*

(v ,wi+1,...,wj,wj+ ,+1,*,* ... ,*]tvi.

this is : [v ,wi+l+l,wi+2,... ,wjwj+1,*,*, ..,*

[v ,w i+l+ ,wi+2,...,wj ,j+l,+l*,*, *]

Which is the same action as tv where

v = w3+1i ; thus (2) is proved.
i+1
For the "only if" part of (1), suppose i/j and v

coincides with the first i entries of wJ; then by what

we just proved: (twj)tvi 3 twj. This implies twj tvi

tvi twj; the proposition is proved.

The structure of the sylow p-subgroup Spn is analyzed

in the next proposition (also see Weir [211):

PROPOSITION 2.7. (1) Spn is generated by all t's

defined in (6.1) and (6.2), consequently,

(6.11) Spn=AOA1A2...An-1 where AO=

(2) S n is a semidirect product of the group T by Spn-l,

i.e.

(6.12) Spn=[T] Spn-1. Furthermore, there is a

linear equivalence relation on Spn such that the equiva-

lence classes have a group structure isomorphic to Spn-l-







(3) If C is the cyclic group of order p, then
p
(6.13) Spn = CplSpn-1 and also

(6.14) Spn = Spn-llCp

PROOF Let x eS n and x={x0,xl,x2,...,xn-_

as defined in (5.9)

(1) For each i=l,2,3,...,n-l and each [v ,v ,...,v.]eQi

the product
xi [Vlv, .-1,v ]
(6.15) p (t [)V
[vi.V2,. .,vi_- ,Vi]
vi=l

is independent of the order of the multiplications in the

vi product by part (1) of proposition 2.6. Here xi[v ,...,

vi]c Q is considered an integer modulo p, since the order

of the element t[vl,...,vil is p. Thus the power (t[ ])xi( )

is well-defined. Consider the element

(6.16) x = P ( P (...(P ((...(t x n (Vl,2,..vn-
n n [v1,2, ... vn-l]
vl=l v2=l Vn-_=l

t[vl,...,vn-21 Xn-2[v', --- n 21])...)

t[vl,v21 x2V 2) t[vl] x( ) t 0

in which the products are executed in the order that
Vn -l=
n-1

first, then followed by T and so on until then
Vn-2=1 Vl=1
finally multiplied by t00.

We assert that x# = x(={x0,x1,...xn-l}), for each

vector W = [wl,W2,...,wn], by 5.9. w x=[wl+x0,W2+xl(wl),

W3+x2(wl,w2),...,Wn+xn ( w1,2,... ,wn-l)]. On the other

hand, w = [wl,...,wn] is fixed by all t's except t0,






t[w, t[ ,w ] ,...t[w l..W hence the image of w by
[Wl] [wlW2] [wly..w- l]
x# is the same as its image by the element.

(6.17) t xn-ll[wl... Wn-11 xn-2
[wl,...wn-1] t[wl,...Wn-21

[wl,...wn-2] x2[lwW2] xl[wl] x0
....t t[ l t
[wlw2] [wl] 0

which obviously maps w = [wl,...,wn] to the element

[wl+X0,w2+xl(wl), ... Wn+xn-1(wl,w2,... ,wn-) )] hence
w x = w x# thus x=x#. This proves that Spn is generated

by all t's. Since Ai is generated by all tv(i)'s with

v(i) so Spn factors into AOA1A2,...An-l.

(2) First we show that TA S n; for each generator

tv(n-1) of T and every xSpn.by (1) x factors into

x=30o~1,.. n-l with aieAi, i=0,1,2,...,n-1, so tv(n-l)x

90 '''I n-l 0 .... n-l D0 1 n-l
t(n-l) v(n-1) t v(n-) v(n-l) (n-1)

Since each 9i is a product of tv(i)'s with v(i)ei by
tv(i)
part 1 and 2 of proposition 2.6, tv(nl-) either equals

tv(n-1) (in which part 1 is applied) or equals tvET for

some vecn_1 (in which part 2 is applied). Thus, t (n-) T
nlv(n-l)
all i hence TAS n. Second, we show that A0A ,.. .An2 is

a subgroup of S n which is isomorphic to Spn-l and is

complementary to T in Spn. The following lemma will show

that A0A ,...An2 is a subgroup of S n:

LEMMA 2.7.1 If G is a group with subgroups S,T such

that S G






PROOF It is trivial, hence, omitted.

Now as a consequence of proposition 2.6 AON(Al),

AOACN(A2),...,AOA1A2,...An-3_N(An-2);thus A0A1, AOAlA2,...

A 0AA2,...An-2 are all subgroups of Spn. For each i=1,2,...

n-1, by the definition of A (6.6) the subgroup A A ,,,Ai

is generated by all t i's where viis an arbitrary element
0 i
in Q. which means that A A ,...A is isomorphic to the

sylow p-subgroup Spi+l of the symmetric group Y i+l on

Qi+l, on the convention that only the first i+l entries

of every vector of on are "moved" by elements in A0A1,...Ai.

Thus,

(6.18) AOA1,...A i Spi+l for each i=l,2,3,...,n-l.

In particular

AOAA2. .An-2 Spn-l and hence

(6.19) Spn= [T] Spn-1 (It is obvious that TnAOA1,...

An-2 = E)

Finally, if we define a relation "%" on Spn by:

(6.20) For each x, yeSpn, define xsy if and only if

x and y induce the same permutation on n-l. (Here Qn-1

can be considered to be the set of all "blocks" of the

form v(n-1) x Q:Q .) Now (5.9) shows that the action of
n
x on the first n-1 entries of vectors is independent of

its action on the nth entry. (Hence, xLx for every xES n.)

It is obvious that "'" is an equivalence relation on Spn

with the linear property

(6.21) If xl1y1 x2%y2 then XlX2%yy2

The equivalence classes Spn/% form a group (called








the quotient group of the equivalence relation) with the

multiplication [x][y]=[xy]. We conclude part (2) of this

proposition by proving the following lemma:

LEMMA 2.7.2 The map e: Spn/T Spn/,. defined by

9(xT) = [x] is an isomorphism.

PROOF (of the Lemma): (i) e is well-defined: for

if xlT = x2T with teT such that xl=tx2, then for each

v(n) = [v(n-l),a] En;

[v(n-l),alx1 = [v(n-1),a]tx2 = [v(n-l),ax2 where a

depends on a,t and v(n-). Thus, xl and x2 induce the

same permutation on all v (n- of n- thus x-2'x

[x ]=[x2] in Spn/%.

(ii) e is one-to-one: if [xl]=[x2] in Spn/n then

Xlx2,a and xI and x2 induce the same permutation on each

v(n-1) n for each a*cE and w(n-1)en let [w(n-)a*]
n-l n-l
-1
= [v(n-1) ,al;then [v(n-l) ,a]x = [w(n-l),a*]. Since

[v(n-1),a]xl and [v(n-l),a]x2 have the same first n-l

entries. So let [v(n- ax2 = [w (n-l),a**];then

t=x1 2: [w(n-) ,a*] [w(n-l),a**] hence teT;thus
xlT=x 2T 0 is one-to-one.

(iii) 0 is obviously an onto map.

(iv) 0 (xT.yT) = 0(xyT) = [xy] = [x] [y] = 6(xT)-* (yT)

(i)-(iv) prove 0 is an isomorphism so the proof of (2) is

completed. The image of any xeSpn in Spn/T= Spn-l is

denoted by x, i.e. x={x0,Xl,...xnl-;then R={x0,x1,...x -2

(3) By (2) Spn=[T]Spn-1 and T=v(nl)n =
n-l
n n-1 vectors(n-1);
(n-1) There are pi vectors v (n
v(n-l) nI v(nl)





'25
hence, there are pn-1 direct summands in the direct

product T, each isomorphic to C the cyclic group of

order p; thus S n can be considered to be the wreath

product CplSpn-1 which is the group of all pairs [see

(4.1)] {(f,b);f:n-_lC p,bESpn-l} in the following way:

If x={x0,xl,... ,xn-} let the pair (f,b) be b=x=

{x0,xl,...,xn-2} and f[(vl,...,vn1))]=xn-1l(vl,2,.. .,vn-) ;

that is, each element x={x0,xl,...,Xn _} corresponds to

the pair (xn-l,x) as defined in (4.1) and (4.2). Hence,

for each vector

(6.22) v(n)=[ (n-l), n]EQn [v(n-l) ,Vn (Xn-1 )=[V(n-1) ,

Vn+xnl(v(n-1))]. If y={y0,Yll...lyn-l then

(6.23) [v(n-1)] (Vn-1,xyn-1y [(n-l) ,v+x (vn-1)]

(n-l') = [v(n-l) y, Vn+xnl(v(n-) )+yn-_(v(n-l)R)]
On the other hand, the composition given in (4.2) is

(6.24) (nl,x) (yn-_1y)= (g,xy) where g = nn-l Cp

is defined by g(v(n-)) = nl(v(n-l) )yn_(v(n-1)) and

hence, its action on v(n) [v(n-l) is

(6.25) [v(n-l),v [g,xy] = [v(n-l) xy,v+g(v (n-l))]
= [v(n-1) -yn+x v(n-l n (n-x)
IV xyv, n-l )+y n-(V

(6.23) and (6.25) are the same; this proves that the

correspondence between x's and the pairs (x n-l,) defined

above is an isomorphism. Hence, Spn=Cp Spn-l.

Notice that it was proved that Spn=Cp Cp ... C ,

n times, which can be viewed either as Cpl(Cpi... Cp) or

as (Cpl... C)ICp)C; the induction gives (Cpl...ICp), n-i

times, as the group Spn-l; thus Spn is isomorphic to both





26

CplSpn-1 and Spn-1lCp- We have shown how Spn is iso-

morphic to CplSpn-l;we are now going to see how Spn is

viewed as the wreath product Spn-llCp:

Consider Spn-1 as the permutation group on the set

n l = {[0,v2"...,vn] Iv v = 2,3,...,n}5nn and the
cyclic group Cp as a permutation group acting on

n* = {[vl,0,...,0]| ViEn~ Qn. Here we use n and n*

instead of n-i and Q to remind the reader what entries

of vectors they act on in order to illustrate the defini-

tion of wreath product given in (4.1), (4.2) and (4.7).

For each x = {x ,x ,....,xn-}, let the corresponding

pair (f,b) in the definition of wreath product Spn-l1Cp

in (4.1) and (4.2) be defined by:

(6.26) b is a permutation on Q, and b:v1 1 Vl+x0

f: n S n-2 is given by f(v) : Qn1 -* n and

(6.27) f(vl) : [v2,v3. .., Vn] [v2+xl (v ) ,v3+x2(v1 v2),...,

Vn+xn+(vlv2,..,vn-l)l

It is a simple algebraic matter to check that with

the definitions (6.26) and (6.27), the composition of

pairs satisfies (4.2) and (4.7);hence, Spn is isomorphic

to the wreath product SpnI-1Cp.



7. Normalizers And Centralizers

In this last section of Chapter II we will discuss

the structures of the normalizers and centralizers of

the two particular p-subgroups of Spn, which control the

local fusion. First, we look at the "big" elementary

abelian group T:









PROPOSITION 2.8 Let T be defined as in (6.7) let

G=ypn, then

(7.1) NG(T)- ([Cp]Cp-l)l Ypn-1

(7.2) NSpn(T) ([Cp]Cp-l)lSpn-1

(7.3) CG(T)= T where C and C are
G p---- ---- p-1
cyclic groups of order p and p-i respectively.

PROOF First note that if geN(T), then g induces a

permutation g on n-i which is the action of g on the

first n-i entries of the vectors. For if there were

vectors [v(n-l),a] and [v(n-l),a+z] such that

[v(n-),a]g = [wl(n-1),a*] and

[v(- ,a+]g = [w2(n a**] with w(n-1) w(n-)

then since (t (n-1)) v(n-1) ,a- [v(n-1),a+(lg

((tv(n-1)) ) : [v (n-1),a]g [v (n-1),a+ ]g

: [wl(n-1),a*] [w2(n-l),a*]

because wlI(n-l) w2(n-l) (t (n-l))g9 T. This contradicts

the assumption that GeN(T).

The conjugation by g of elements in T is an auto-

morphism of T; thus it maps a generating set to a generating

set. Since T is generated by all tv(n-1) with v(n-1)

ranging over all vectors in n-l' the conjugation by g

permutes the pn-1 cyclic subgroups t (n-1) .Since the

set of permutations on these pn-1 cyclic subgroups is iso-

morphic to Ypn-l1 it constitutes the "top" group of the

wreath product. For the notation of top group see definition

2.2. Now let BeYpn-1 be the element of the top group








corresponding to g; for each v=v(n- En let w=w

be the vector in _n-i such that ()g =, that

is, vB=w. Now we have the p-cycles tv=t (n-)

([v,l] [v,2]...[v,p]) and tw=tw~n-l) = ([w,l] [w,2]

...[w,p]). The bottom group of the wreath product

contains the set of maps.

(7.4) f: { [v,l] [v,2] ,...,[v,p] } { [w,l] [w,2],...,

[w,pl} satisfying

(7.5) (t )fa=(tw)a for some a {1l,2,3,...,p-l}

Here fa is one such map which corresponds to aEQ. Then

in the cycles tv=tv(n-1)=([v,l][v,2],...,[v,p]) and

(tw)a=([w,l][w,l+a [w,l+2a],...,[w,l+(p+l)a]). far's
must match the vectors in their order but free of choice

for the first vector in the cycle. Thus, for each bE{0,1,
b
2,...,p}there corresponds a map fa which maps [v,vn] to

[w,l+(b+vn-l)a] for all n=l1,2,...,p; i.e.

fb: [v,v ]- [w,vn.a + (ab-a+l)]. For each fixed a,

{ab-a+1}b is the same set as Q itself. (b-+ab-a+l is

a linear function of b). Thus, as b ranges over all

elements in Q, fa ranges over all maps of the form

(7.6) fb: v,vn] [w,vn.a+b].

The set of all such maps fb can be realized as the

group of all Affine transformations on the Affine line.

This group is the semidirect product of Cp by Cp_- where

Cp is the set of all fb's and Cp-1 is the set of all far's.

Now this group [Cp]Cp_- is the "bottom" group of the

wreath product because the above definition of [Cp]Cp-1







comes from every v=v(n-1) cn-l and the pairs (f,8)

with c-Ypn-i and f= an2 [cp]Cp-l is f:v(n-l) fba

defined above satisfy (4.1) and (4.2) of the definition

of wreath product. Hence, NG(T)= ([Cp]Cp-l)1 ypn-l.

It is easy to see (7.2) comes from (7.1), to show
g
(7.3). Let g be an element in C(T); then t (n-l)=t (n-1)
v v
for all v(n-l) n-. For each v(n-1) and all i=l,2,...,p,

the permutations:

Zvtn-1) [v(n-1),vn] [v(n-l) ,n+i] all v nQ
Ziv(n-l)
all satisfy t = t (n-1) and every permutation
v(n-l1) v
on which satisfy this equation comes from one

such Z ; thus the group H (l, which
v(n-1) v(n-1)e n v(n-)
n-1
is isomorphic to T, is the centralizer of T in G. The

proposition is proved.

Recall that for each v(n-2) (=v)e
n-2
(7.7) t = E ([v,a,l] [v,a,2],... [v,a,p]) is a product
v aEQ
of p p-cycles and

(7.8) tv= e ([v,l,b] [v,2,b], .. .[v,p,b]) is also

a product of p p-cycles. We now discuss the structures

of the normalizer and centralizer of the group generated

by t* and tv:

PROPOSITION 2.9 For each v=v(n-2) :
n-2
(7.9) N() = ([Cp]Cp-l)x([Cp]Cp-l))x v

(7.10) C() =(CpxCp)xy, where Yis the symmetric

group of degree pn_ 2 on the set ~n-{v}x2

PROOF It is obvious that every permutation of n





30

leaving all p2 vectors in {v}xo2 fixed lies in the normalizer

and centralizer of . Hence, the factor Yjhas

little significance as far as the structure is concerned.

Thus, through the proof of this proposition we will use

t to denote t and use t* to denote t* and [a,b] to denote

[v,a,b]; hence, we are looking at the symmetric group Yp2

on the set Q2. We prove (7.10) first:

(7.11) t* = H ([a,l] [a,2],...,[a,p])
bEn
(7.12) t = ([1,b] [2,b],...,[p,b])

If geC( ) then tg=t and t*=t*. From t*=t *

g permutes the p p-cycles {[a,l] [a,2],...,[a,p] };hence, g

induces a permutation lJ on the first entries; from

tg=t, g permutes the p p-cycles {[l,b] [2,b],...,[p,b]};

hence, g induces a permutation 2 on the second entries;

thus, g:[a,b]-* [agl,bg2] all a,b. Hence,

(7.13) ([a,l][a,2],...,[a,p])9=([agl, g2 [ [ag,2g2] ,. .

[ag,pg2}) = ([ag,l][agl,2][ag1,2],...,[agl,p]). Hence,

there is j such that the corresponding

(7.14) gj:[a,b] [agl,b+j] all a,b, likewise,

([l,b [2,b] ,...,[p,b])9 = ( [1gl,bg2] [Igl,bg2] ,. .,[pgl,bg2

= ( [l,bg2l [2,bg2] .,[p,bg2] )

Hence, there is i such that the corresponding map

(7.15) g ;[a,b] [-i+i,bg2]. Hence, for each pair

ii
i,j, there corresponds a map g. : [a,b] -+ [a+i,b+j] These

g 's are all the permutations in the centralizer of .

The set of all get's, where i,j=0,1,2,..,p-l, can be

realized as the direct product Cp x Cp which is isomorphic

to thus C() = = CpxCp.





31

Next we show N()= ([Cp]Cp-l)x([Cp]Cp-l).

If gcN() then there exist i,je{0,1,2,..,p-l} such

that

(7.16) (t*)9=t*it : [a,b] -[a+i,b+j] all a,be .

We will find all g such that 7.16 holds and tge ,

since t* = ([a,l] [a,2],...,[a,p])
aEQ
(t*)9 = ([a+i,l+j] [a+i,2+j],...,[a+i,p+j]);thus g
aES
can be found to send the cycle ([a,l] [a,2],...,[a,p])

to any of the p p-cycles in (t*). Let cyp be a permuta-

tion on Q which sends the vectors in the cycle ([a,l],...,

[a,p]) to the cycle ([a +i,l+j][a +i,2+j ,...,[a3+i,p+j]).

Let g; be the element of Sp2 which corresponds to such 3.

Now, for each and each a, there are p ways to.match

the vectors {[a,l],[a,2],...,[a,p]} and the vectors

{ [a+i,l+j] [a+i,2+j],...,[aZ+i,p+j]) ;i.e. there exists

a9- depending on a such that

(7.17) gD : [a,b] i[a +i,b+j+ak]

Since t = ([l,b] [2,b],...,[p,b]) and tg must lie in

, say tg=t*j ti for some integer i,j modulo p, then

(7.18) t : [a,b] [a+i ,b+j'] which coincides with

Ibe ([1 +i,b+j+l (2 +i,b+j+2z] ,....[p +i,p+j+p ] ) which

reads as [a +i,b+j+a ] -[(a+1) +i,b+j+(a+l)']. Comparing

(7.18) and (7.19) we have

(7.20) (a+l) a9= i and

(7.21) (a+l)'- ak= j

Thus (7.20) and (7.21) are necessary conditions which

define the elements geN(); these limit the ways








of assigning the cycles and in each cycle, the ways of

"shifting" the vectors. Note that the definition (7.17)

of gZ doesn't depend on i,j because as a,b range over

all Q the pair [a +i,b+j+a ] also ranges over all n2'

Hence, the definitions of D and k (both permutations

on s) define the element gEN( ). It is easy

to see the definition of a and k satisfying (7.20) and

(7.21) is equivalent to the definition of the direct

product ([Cp]Cp-l)x([Cp]Cp-l) of pairs (9,k) where

a:Q -Q, ;:R such that D(a)=ha+m all aER and

k(a)=Ka+m' all acE where h,K E {l,2,3,...,p-l} and

m,m' E{0,1,2,...,p}. Hence, N() ([Cp]Cp-l)x( [Cp]Cp-l)

the proof of proposition 2.9 is completed.












CHAPTER III
FUSION OF ELEMENTS
IN Spn I



8. Cycle Structure and Conjugacy Classes

Each element x of Spn can be written as a product

of disjoint cycles. If we consider vectors fixed by x

as cycles of length l=po, then all cycles which appear

in the cycle decomposition of x are p-power cycles: the

maximum cycle length is the order of x. Since Inl= pn

the order of x is at most pn and the sum of the lengths
n
of all cycles (including po-cycles) equals p Thus,

in Spn, each partition of pn into p-power numbers gives

a cycle decomposition,and since two elements of Spn

are conjugate if and only if they have the same cycle

decomposition, each cycle decomposition represents a

conjugacy class in Spn. Thus, there are as many conjugacy

classes in Spn as the number of partitions of the integer

pn into p-power integers.

If x={x0,xl,x2,...,xn-l}eSpn and if x00, then x

fixes no vectors of n If x =0 then x is a product of
n o
p elements l'2"... '' p where for each i, Hi is a permu-

tation acting on the subset {i} x Qn-l of an, hence,

an element of Spn-l. If xcT, then x0,xl,x2,...xn-2 are

all zero maps and x is a product of as many p-cycles

as the number of vectors v(n-l) such that xn-l(v(n-l)) 0.







For each v() E, j=l,2,...,n-l, tv(j) and t*(j) are both

products of pn-j-1 p-cycles, while tO and t are products
n-1
of pn-1 p-cycles.



9. The Main Theorem

For each xeSpn, let x be the image of x under the

canonical homomorphism:Spn Spn/T Spnl;i.e. x is a

permutation of degree pn-1 induced by x on the first n-l

entries of the vectors of n R can be called the action

of x on the "blocks" which are subsets of Qn of the form

{v(n-l) }x where v(n-1) ranges over Pn-l. Define x as

the permutation on Qn-2 induced by x in the same way.

Although there are a large number of conjugacy classes,

the fusion of elements in Spn is surprisingly simple; as

we will prove soon, only two kinds of local subgroups are

needed to control the local fusion of Spn. They are

the normalizer of the big elementary abelian group T,

and the normalizers of some small elementary abelian groups
2
of order p This is stated in the following theorem:

MAIN THEOREM: The following two families of local

groups control the local fusion of Sn :

(9.1) F1 = {N(T),N(); v(n2)ranges

over n- }
n-2
(9.2) F2 = {N(T),N(); where the products
2 v v V v
range over any subset (the same set for both products)

of n-_2.

We will prove the main theorem in two parts: Theorem 1





35

proves the fusion of elements x and y in S n with x and

y conjugate and R and y are also conjugate. Theorem 2

proves the general case.

THEOREM 1: If x,ye Spn are conjugate and if x and

Share also conjugate, then there is an element g in

the normalizer of T such that x =y. In other words,

N(T) controls the local fusion of elements in Spn having

the same cycle structure on "blocks".

PROOF Let x = {x ,x,x2,...,xn-_} and y = {y0,YlY2,..

,yn- } as defined in (5.9). For each vcQn, let {vx}
denote the orbit of v under x; thus {vx} is the set of

all vectors in the cycle of x containing the vector v.

Let j{vx}I denote the cycle length. For each v(n-l) E Qn-i
and each v(n-2) n_2,the orbits {v(n-l)} and {v(n-2)

and their cycle lengths are defined in the same way.

First, we have to prove a lemma concerning the relation

between the cycle lengths of vx and v(n-l)x when v=[v(n-l),a]

for some aQ :

LEMMA 3.1 Let XESpnfor each v(n-l) n1 and each
(n-p1) n-
ae if {v x}l =p for dro, then either

1{[v ,a]x = p or
(n-1) d+l
I{[v ,a]xl = pd

Furthermore, if I{[v(n-l),a]xl = p for one aes
>(n-1)
where y-o, then {[v(n-l),b]xl = p for all bse
(n-l)- d
PROOF Assume I(v n x}l = p Define

(9.3) 6x(v(n-l) = Xnl (w(n-))

w(n-11 {v(n-l) }







For each aeo
(n-1) = [v(n-1)- (n-1)
[v n- a]x = [v x, a+xn (v-l))

(v (n ,a]x [v(n-)2,a+xnl(v(n-))+xnl(v(n-x)


(9.4) [v(n-l),a]xpd = [v(n-l) a+x(v(n-l))] because of

v(n-l)xpd = v(n-l) and the definition of 6x(v(n-l.

Case (i) If 6x(v(n-1))=O (modulo p),then for any aEn,

[v(n-1),a]xpd = [v(n-1),a] by (9.4) and pd is the smallest

power of x which fixes the vector [v(n-l),a]. Hence,

{[v(n-l),a]x} = pd and in the orbit {[v(n-l),a]x} all

the vectors v(n-l) 's are different so there is no b/a

in Q such that [v(n-1),ble {[v(n-l),a]x}. So, {[v(n-),b]x}

b=l,2,...,p are p disjoint orbits of length pd. We call

a pd-cycle of x of this type a type I pd-cycle of x.

Case (ii) If 6x(v(n-1)/0 (modulo p),then

v(n-,axpd = [v(n-l),a+x(v(n-l))] by (9.4)

[v(n-1),aix2P [v(n-l),a+2.-x(v(n-1))]
*
(9.5) [v(n-l) ,a]xipd = [v(n-1) ,a+i-6x(v (n-1)

Since a+i-6x(v(n-1)) ranges over all vectors in Q as

i ranges over Q {[v(n-l),a+i-6x(v(n-l))}P is
i=1
equal to {[v(n-1),a]} Hence, [v(n-) ,a]xPd+l
aeQ
[v (n-),a] and d+l is the least power of x which fixes
S-l ,a] (n-1) d+l
[v(n)a]. Hence, {[v ,a]x}I = p and the

orbit {[v(n-l),a]x} contains [v(n-1),b] for all be
(n-i) d+l
Hence, {[v n ,b]xl = p for all ben We call

pd+l-cycles of x of this type a type II pd+l-cycle

of x. Thus, the lemma is proved.







Now since x and y are conjugate in ypn, their

cycle structures are the same. That is, they have

the same number of pr-cycles for every r=0,1,2,...,n.

Furthermore, since R and y are conjugate in Ypn-l1

x and y also have the same number of pr-cycles for

every r=0,1,2,...,n-1. The following lemma is the key

to the proof of this theorem:

LEMMA 3.2 If x,yESpn are conjugate and if x and y

are also conjugate, then for each r=0,1,2,...,n the

number of type I p -cycles of x equals the number of
r r
type I p -cycles of y, and the number of type II p -cycles

of x and y are equal.

PROOF We prove the lemma by induction. For each

r=0,1,2,...,n let X1 (resp. X2) be the number of type I
r r r
(resp. type II) p -cycles of x. Let Y1 (resp. Y2) be
1 2
the number of type I (resp. type II). p -cycles of y
r
and let Xr,yr be the number of p -cycles of x and y

respectively. (Here Xr=Yr because x and y are conjugate.)

Then, Xr= r+ 2r, yr=y ryr, for r=0, X1Y are

numbers of fixed points of x and y respectively and

X20=Y2 =0 because there is no type II p -cycle. So,

obviously X 0=Y 0=XO=YO. Let's now look further at the
1 1
case r=l; X1 = the number of type I p-cycles of x which

look like ([v(n-l)a (n-l) (n-),a ) for
,aI] [v ,a2]1,... [v ,ap)
(n-l)
some v e n l and {a ,...,a }= 0 Each such v(nl)
n-( n p
has the property that Xn-l(v n-) 0 (modulo p); thus

(9.6) X2 = the number of fixed points v(n-1) of x

with xnl(v(n-l)) / 0 (modulo p).







On the other hand, if w(n-1)E n- is a fixed

point of x with xnl(w(n-1)) 0 (modulo p),then

[w(n-),a] is a fixed point of x for every aEQ This

means each such w(n-1) gives p fixed.points of x. Since
-0 1 (n-l) -where o
there are (X -X) such w ,where x is the number
2
of fixed points of x, we have the

(9.7) relation: p.(X-Xl)=X= the number of fixed
2
points of x, the same equation holds for y;

(9.8) p.(Y -Yo)=O= the number of fixed points of

y.

Now by the assumption, XO=YO, XY, the above two

equations give the equality
1 1 1 1
X2 = Y ; hence, X Y1

This proves the case for r=l.

Now assume the lemma holds for r=d. That is

(9.9) x d = d, x2d=2 d

Since for each r, XrY, Xr=yr, we use Cr to denote X ,r,
fr each r,
and use Cr to denote X Y just to note they are constants

because x and y are given to be conjugate and so are

x and y.

If {v(n-)x} is an orbit of x of length pd and if

6x(v(n-1)) (mod p) then v (n-) gives exactly p type I

p -cycles of x; they are (use v=v(-1)) ([v,a] [vx,a+xnl(v)]

.... [v Pd-l,a+***,...]) a=1,2,3,...,p.

All type I pd-cycles of x arise in such a manner,
-d d
(9.10) let ml be the number of p -cycles of x which

give type I pd-cycles of x, then






d -d
(9.11) x = pml Now
(1-d d
(9.12) Cd-m = the number of p -cycles of x which give

type II p -cycles of x. Since every type II p -cycle

of x comes from exactly one pd-cycle of x and every such

p -cycle of x gives exactly one type II p d+-cycle of

x, hence, we have

(9.13) C m = Xd+l
1 2
Combining (9.11) and (9.13) we have

(9.14) Cd_ Xld = X2d+l
-p2
the same equation holds for y; we have

(9.15) Cd_ Y Yd+l
P
From (9.14) and (9.15) together with the inductive hypoth-

esis X 1Y we have

(9.16) X2d+l = Yd+1 hence, also

(9.17) X1d+ = X1d+

Thus, by induction, X and X2 r for all r=,2,3,...
rand X =Y for all r=1,2,3,...

The lemma is proved.

Now for each d=0,1,2,...,n and each p -cycle of x,

we can find a corresponding pd-cycle of y of the same type

because of Lemma 3.2. So there is a one-to-one correspon-

dence between all cycles of x and all cycles of y pre-

serving the length and the type. If we choose a permuta-

tion g which maps vectors of each cycle of x into vectors

in the corresponding cycle of y in such a way that the

order the vectors appear in each cycle is preserved;

that is, if the corresponding cycles of x and y are

written:





40

(9.18) (v,vx,vx2,...,v d-1) and (w,wy,wy2,...,wypd-i)

then g maps vector vxi to wx for every i=p,1,2,...,p ,

then obviously xg=y if such g is defined. The final

step is to show that some such g defined by an appro-

priate expressions of cycles of x and their correspond-

ing cycles of y lies in the normalizer of T.

For each v=v(n-l) n-, let {v x} =pd, we con-
n-i
sider two cases at length

Case 1: Suppose{vx} gives type I p -cycles of x. Let

the corresponding pd-cycle of y be given by the orbit

{wy} of y where w=w(n-l) n-l. Because of Lemma 3.1,

the p vectors [v,a] a=l,2,...,p all lie in different

orbits of x and the p vectors [w,a], a=l,2,...,p also

lie in different orbits of y. We now choose the one-

to-one correspondence such that the orbit {[v,a]x}

corresponds to the orbit {[w,a]y} for each a=l,2,...,p.

Then we define the element g to send [v,a]x to

[w,a]y for all a=l,2,...,p and all i=0,1,2,...,p -1,

this way the element g is not defined only on the

vectors [v,a] a=l,2,...,p but also on the vectors
i dd
[w,a]x for all i=o,l,2,...,p -1. In other words, there

are p pd-cycles of x and y involved in this definition

of the element g. We will now show that for each
v-i pd
vx of these p vectors {vx }pd1 we have t ieT.
1=o vx
-iO
For every vector vxi in the orbit {vx}of x, the

vector given by vxi which lies in the orbit {[v,a]x}

of x is





41
i-1i
(9.19) [vx ,a+ E x (vxJ)] and the corresponding
j=o n-i
vector of y is
i-1
(9.20) [wy ,a+ E y (wy)] let
j=o n-1
3=0
i-i i-i
(9.21) S = x (v-j) and p= i y -(wyJ)
j=o n-1 j=o

since t vi:[vRi,a+si] l- [vxi,a+l+si]
vx i 1
(tvgd:[vxia+sig 19 [vxi,a+l+si]g

(9.22) :[wya+pi] [wy (a+l)+pi]

Since a+pi ranges over Q as a ranges over Q,(9.22)

means (t i)g = twi e T.
vx wy
Since type I pd-cycles of x all come from some
d d
p -cycles of x and each such p -cycle of x gives p type

I pd-cycles of x, the number of type I pd-cycles is a

multiple of p for each d=0,1,2,...,n. So the above

defining process of the element g is repeated until

all type I cycles are exhausted. So, (tv(n-l) ET for

all such v(n+l)EQn-1. Thus, we have defined g on all

v(n-l). n_1 which lie in some type I pd-cycle of x and

have shown (t (n-l))ET for all such v(n-1); the next

step is to define g on all other vectors v(n-l), that

is, vectors v(n-1) which lie in some type II pd+l-cycle

of x.

Case 2: Suppose {vx} gives a type II pd+-cycle of x;

then Sx=6x(v) / 0 (modulo p). Let {wy} be the orbit

of y which gives the pd+-cycle of y which corresponds

to the type II pd+l-cycle of x containing {vx}. And

let 6y=6 (w) / 0 (mod p). Note that the orbit {[v,a]x}

contains all p vectors of the form [v,b] b=l,2,3,...,p.







So does the orbit {[w,a]y} contain all p vectors of

the form [w,b],b=1,2,...,p. Let's write these two

(type II) pd+l-cycles so that [v,l] and [w,l] are

both the first vectors in the cycles. Then define

g on these pd+l vectors by sending each vector in the

cycle {[v,llx} to the corresponding vector in the cycle

{[w,l]y} That is

(9.23) g:[v'i,l+si] [wyi,1+pi],i=0,,2,.... d+1-1.

Let q=pd; then for each is{0,1,2,...,qp-l} there exist

k,me{0,1,2,...,p-l} such that

(9.24) i=Zq+m.

Since i{vx}j=pd=q, vxi=vxq+m = vxm and hence,
i -
(9.25) Si= E x (vx3) = .6 + S by the definition
1 j=o n-1 x m
of 6 in (9.3). Hence,
x
(9.26) [vx ,1+S.] = [vxm,l+ .6 +S ]
1 X m
likewise

(9.27) wy = wy and Pi=z.6y + so,

(9.28) [wyil+pi=[wym, 1+A.6 y+pm] and

(9.29) g:[vxm,l+k.6x+S,] [wyml+Q.6y+pm] for all

m,Z

For each fixed m=0,1,2,...,p-l. Consider the p
-i
vectors {[vx,l+s.i] i=zq+m ==0,1,2,...,p-l} because
-m
of (9.26). This set equals {[vxml+..6x+S ] I=0,1,2,...p-1}

and the corresponding vectors in y are {[wyml+Z.6 +p

S=0,lO 2,...,p-l}

(9.30) g: [vxml+.6 x+S ] i [wyml+t.y +nm for all

Z=0,1,2,...p-l. And because







tvxm: [vm 1+,6x+Sm] [vxm 2+,6x+S m

(tvxm)6x :[vm 1+i+ .6x+Sm,] [vxm, 1+6 + Z.6x+Sm]

((tvxm)6 9; [vxm, 1+z .6x+Sm]g [vm, l+(z+1)6x+Smg

that is, ((t m)6X)g : [wm l+6y+m]

[wym +(k+1)6 y+m] because of (9.3) for all

9=0,1,2,... ,p-l.

Since (1+(Z+1)6 +i )-(l+z6 +m )= 6y and since

a=l+L.6 y+pm ranges over all elements of Q as & ranges
6x
over 0,1,2,...,p-l. This means ((tvym) )g :[wim, a] -

[wym, a+6 ] which is the same action as (twym) 6; hence

we have

(9.31) ((tvm)6x )9 = (t m) 6. This will imply

(tv-m)geT with the help of the following lemma:

LEMMA 3.3 If the integers h,k / 0 (modulo p)

exist such that

((t ) ) = t where v,w,eE1 and
v w n-l
gey then (tv )9 T.

PROOF As h and p are relative prime there are

integers e,ff0 (modulo p) such that he + pf = 1

t g = (t vl) = (tvhe+pf)g = t heg t pfg

Stvheg E (because tvP = E)

= (((t )h)g)e E

(t k) e
ke
= t i:T
k
In fact, ke = because hesl (modulo p);the lemma is

proved. Hence, the proof of Theorem 1 is completed.

We conclude the chapter by showing the following

proposition which we will need in Chapter 4:







PROPOSITION 3.4 If x,yeS n such that for each

d=0.1.2,...,n, the number of type I pd-cycle of x and

y are the same and the number of type II p -cycles of

x and y are the same, then x is conjugate to y in y

and x is conjugate to y in ypn-

PROOF Equations (9.14) and (9.15) would become

(9.32) Xd Xld = X d+l and
P 2

(9.33) d Yd = Yd+
-d -d
if we did not assume X = Y Now, the assumptions

dd and d+1 d+l id -d
Xld = Yld and X2d = Yxd for all d imply d = d
d d
by (9.32) and (9.33) and it is trivial that X Y So

x and y have the same cycle structure and hence, conju-

gate. Also, x and y have the same cycle structure,

hence conjugate in Spn-l; the proposition is proved.













CHAPTER IV
FUSION OF ELEMENTS
IN Spn II



In this chapter we will prove the second half of the

main theorem, namely find the local fusion family which con-

tros the fusion of the entire Spn including conjugate elements

with different cycle structures on "blocks". We will see

also how fusion of direct product is related to the fusion

of its direct summands, thus, see in general the fusion

of elements in the Sylow p-subgroups of symmetric groups

of any degree.



10. The Second Theorem

Theorem 1 shows that the conjugation of two elements

with the same cycle structure on "blocks" can be taken

from the normalizer of the subgroup T; this means N(T)

provides all local fusion for elements which lie in the

same "subclass" of a conjugacy class; here a subclass is

a subset of a conjugacy class containing all elements in

this conjugacy class which have the same cycle structure

on blocks. More precisely, for each xeS n, let [x] be

the conjugacy class of Spn containing x; i.e. [x]={yESpnlxg=y

for some geYpn} ; and let [x]={yESpnlx and y are conjugate

in Ypn-l} (note here [x] is not the conjugacy class of




46

Spn-1). Theorem 1 shows that N(T) gives the local fusion

of all elements in a subclass [x]f[x]. See figure 4.1.















Figure 4.1



The remaining of the main theorem is to find a local

conjugation family (or families) which controls) the

fusion of elements in (possibly) different subclasses.

See Figure 4.2.












Figure 4.2


In Theorem 2, given such elements x,y we will find

elements x*,y* in the same conjugacy class such that x

is conjugate to x*and y is conjugate to y* (thus by Theorem

1, both conjugations are in N(T)), and that x*,y* so found

depend only upon the number of cycles of all lengths and

types and prove that x* is locally conjugate to y* in

the families F1 and F2 of (9.1) and (9.2):







THEOREM 2. If x,yeSpn are conjugate then there

exist gl9,g2..*.gr such that xg9lg2 gr=y and all gi

lie in members of each of the following two families:

F1={N(T) ; N() v(n-1) ranges

over n2 }
n-2
F2={N(T) ; N(< lIt* n tv>) : where the products
V v ,V
range over any subset (the same set for both

products) of n-2

PROOF First we prove some lemmas which are needed:

LEMMA 4.1. For each d=l,2,3,...,n and any vector

w=w(n-d-1) en-d-1 letA ={w}x _d-1
exists a pd-lcycle zeSpn-2 such that all vectors of

Qn-2 outside A are fixed by z.

PROOF (of Lemma 4.1): Let z={z0,zl,...,zn-3} with

z0=0 and Zl,Z2,...,zn-d-2 are zero maps, Zn-d-l=Cw'

Zn-d=~[w,pl' Zn-d+l = [w,p,p]''''' n-3 = [w,p,p,...,p]'
d-2
where S's are defined in (6.1) then it is easy to see

that z is a pd-l-cycle which leaves all vectors outside

A fixed.

LEMMA 4.2. Let A and z be defined as in lemma 4.1

and let v be any vector in A let r=pd, define the

elements z1 and z2 as follows:
r-1
(10.1) z = b ([v,l,bb][vz, ] ... [v -l, ,b] lv,2,b]
[vz,2,b ...[vzr-l r-l
[vz,2,b]... [vzr-1 2,b]...[v,p,b] [vz,p,b]...[vzr-1 ,p,b])
p r-l
(10.2) H2= 6 ([v,a,l] [vz,a,l] ... [vzr-1,a,] [v,a,2]
-a=l
vz,a,2]...[v r-l ,a,2 ...[v,a,p][vza,p] [vzr- ,a,p]
fvz~a2] .. [v a,2 ...[vra,P] [VZiaapl...[vz ,a,p])







then (1) zl is a product of p type I p -cycles while z2

is a product of p type II pd-cycles on AxxnQ.

(2) Let gey such that g fixes all vectors outside
pn
AxnxQ and in Ax.xQ let

(10.3) g:[vzi,a,b] [vz ,b,a] for all i=0,1,2,...,
r=d- and all aF., beQ.

(10.4) then (a) z g = z2 z2 = z1 (hence g2 C()
r r
(10.5) (b) gEN(< l t i >)
i=l vz i=l VZi
(c) g factors into g=gl9g2. .gr such that
d-l
(10.6) gicN() for each i=1,2,...,p l=r,
X; d
PROOF (of lemma 4.2): (1) Since {[v,l] z11}= p

and j{[v,l,b] zl}= pd for every b, thus each b gives a

pd-cycle of zl which is type I. So zl is a product of

p type I pd-cycles. While for each a=1,2,...,p,{ [v,a]w } =

pd-1 but { [v,a,l]z2 }= pd,thus each a gives a type II

p -cycle of z2;hence z2 is a product of p type II pd-cycle.

(2) By the definition of the element g and part (3) of

proposition 2.1, zi9 = z2 and z29 = zl;hence (zg) =

22g = 21 and ((22)g) = (21)9 = z2 g2 C(zl) and g2EC(2)

g2EC() 2(a) is proved. We now prove 2(c) first:

for each i=1,2,3,...,r=p -, let gi be defined on the

set {[vzi,a,b] aEQ,be2}={vzi}x Q x by

(10.7) g: [vzi,a,'b] [vzilb,a] all a,b. While leaves

all vectors outside {vz }nQx x fixed; then obviously

g=gg92...gr and since tvzi:[vzi,a,b] -. (vzia+l,b] for
all a,b (tzi)gi: [vzi,a,b]gi [vzi,a+l,b]gi ;

i.e. (tvzi)9i:[vzi,b,a,] [vzi,b,a+l] all a,b by the








definition if gi, but this is the same as the action of

tvzi* (see definition 6.4). Hence,

(10.8) (tvzi)i = tvzi*

likewise (tvzi*)i = tvzi hence

gie N() 2(c) is proved.

For part (b) because

r tvzi:[vzi,a,b] + [vzia+l,b] all i; a,b.
i=l

S t )g: [vzi,a,b]g [vz ,a+l,b]g all i; a,b.
i=l v

(11 tvzi)g:[vz ,b,a] [vzi,b,a+l] all i; a,b.

r
this is the same as the action of igltvzi*
r g r
(10.9) So, ( t i) = t i
= vz i=1 vz
r g r
similarly ( itvi*) i= t i;hence, () is proved.
i=l zi i=l vz

Now to prove the theorem, given two conjugate elements

x and y in Spn, we will find two elements x* and y*eSpn

such that (i) x is conjugate to x*,(ii) x is conjugate to

x*, (iii) y is conjugate to y*,(iv) y is conjugate to y*

and (v) x*,y* depend only upon the number of cycles of

x and y of all lengths and types; that is, x* depends

only on the numbers x ,x ,xd for all d=0,1,2,...,n, and
d d d
y* depends only on y ,yly2 for all d=0,1,2,. ..,n, (see

proof of lemma 3.2 for the notations). And the most

important of all, (vi) x* and y* are locally conjugate

in the families F1 and F2 defined in (9.1),(9.2).

(10.10) For each i=l,2,...,n, the number of all vectors

in cycles of x of lengths less than p is a multiple of pi,






i
let it be 6..p We first assume that for every i,

16.
El, j times. The assumption 156.


6n=1 and x,y are products of cycles of lengths less than

or equal pn- and x,y own pi--cycles for i=1,2,...,n.

So obviously x,y are not pn-cycles. However, if both

x,y are p -cycles, then they are both type II pn-cycles.

Hence by theorem 1 they are conjugate in N(T). For

general case beyond the condition 16i.


is similar to what we are going to do below, hence, is

omitted.

Define an increasing sequence of subsets of n as

follows:

A = E x E x x E x E x x x= En-2 x A1 xQ

2= E x Ex .x E x x x = En3 x A2 xn2



(10.11)

Ad= E x E x xExAdx x x x .x Q = En-d1 x x Ad d


n-l= An1 xnxQx ... .x = An1 xn-l

A =n
n n

We will define the permutation x*,y* using A. as
1
vectors in cycles of x* and y* of lengths less than p

in such a way that the corresponding cycles of x and x*

are of the same type and so are the corresponding cycles

of y and y*.

For each d=1,2,...,n-2. If a p -cycle of x induces





51

a type I pd-cycle of x,then it induces exactly p type I
d d
p -cycles of x (see lemma 3.1). That is, xl is divisible

by p. Let x = p.m (this m is the same as md in (9.10)

and (9.11)). Now the total number of vectors in p -cycles

of x is

(10.12) pd Xd = 6d+lPd+ d- d (by the definition (10.10)

of 6d')

= (p-6d).pd + (6d+l-1).pd+
dd d d
= p .x1 + p .x2
d x d d
=p :.(P.md ) + p.x2

= pd+l.m + p.x hence,
dd d+l
(10.13) pdx (p-d).p = ((6d+l-)-mx) pd+1 0 (modulo

pd+l) likewise, for y we have,
dd d d+l
(10.14) pd y (p- d = ((6a+ll)-m ).Pd1 0 (modulo

d+l)
d d
(10.15) hence, x2 y2 P-5d (mod p). Let

(10.16) d = ExEx ...xEx(n-Ad) xxox...xQ then
(n-d-l)times d times

(10.17) d= (p-6d)

then the number of vectors contained in type 7Ipd-cycles
d+l
of x less |0a will be a multiple of p +(by (10.13),

and the same for y by (10.14). Define

(10.18) = ExEx...xEx(A d+-E)xxnxx ...xn then
(n-d-2)times (d+l)times

(10.19) ed[ = (d+l-1).pd+

Note that both sets d and ed work simultaneously for

both x and y because they depend only on 6d's and Ad's

which are the same for both x and y. It is easy to see:







(i) d and ed are disjoint

(ii) a+1 Ad = i d

(iii) If we define Jn-_ = (-An-_)XQn-_ = An-An- then
n-i n-2
n = An = U Id d=l ed disjoint union.
n n d=l d=1 d

We will use the vectors in Td as vectors in some
d d

(p-6d) typell p -cycles of x* and y* and use (6a+l-l-mdx) d
vectors of ed as other (6d+l-l-mdx) type II pd-cycles

of x*, then the last mdx vectors of 8d as mdx type I

pd-cycles of x*. Likewise, (6d+l-1-md pd vectors of ed

will be used to form type II pd-cycles of y* other than

the (p-6d) mentioned above, and the remaining md.p vectors

of 8d will be used to form all type I p -cycles of y*.

See figure 4.3


6d+l


d
dP
for type II p -
_ycles of x*

(P-6d) ^.p
vectors for
type II p -cycles of x*





vectors for type I pd-cycles
of x*


-------- 3^~~- ~~ --- '
d+l
p


6d+l-l-m





md
d


Figure 4.3


Now write x,y as products of cycles in an order that

cycles of smaller lengths first and in cycles of the same





53

length, type II cycles first, namely the order is: fixed
2
points, type II cycles, type I p-cycles, type II p -cycles,
2
type I p -cycles, and so on. This way we get a one-to-one

correspondence between cycles of x and y preserving the

cycle length. Although this correspondence does not

preserve the type but for each d, the first (p-6d) p -cycles

of both x and y have been ordered that they are all type II.

By (10.15) the number of pd-cycles which x,y are of

different type is a multiple of p namely, (m$ -mdY).p.

For each d=l,2,...,n-1, define xd and y* on U8d

Ad+-A (let 8 = n) inductively as follows:
d+l d n-l

x ,y fix all vectors in A On = E x(Q-A)xl both
S1 1 1 n-2x 1
*
x and yl map E x[a,b] to E x[a,b+l] for all aEP-A,
1 1 n-2 n-2
*
ben. Thus, xI yl act on 1l as products of (p-61)
2 x =
type II p-cycles. Let B = {2,3,...,62-m} and C 2

A2-B2x-1;then IB2I = 62-1-m IC2Xl = mlx and define
2 21 2 1 2
B2y,C2y similarly. Then, let xl map E n-3x[a,b,c] to

En_3X[a,b,c+l] for all b,cEQ and all a. B2 and map

En_3x[a,b,c] to En-3x[a,b+l,c] for all b,ce and all

as C2x. Thus, x is defined on A2 so that the number

of type I (type II) p-cycles is the same as that of x.
*
The definition of yl on A2 is similar. Obviously xl
*
and yl are conjugate.

Inductively, assume x and y are defined on
d-1 d-1

d-1U ed-1 = Ad-Ad-l;both are products of p -cycles
and are conjugate and Xdl contains as many type I
d+l
p -cycles as x does; y contains as many type I
d-1





54

pd+l-cycles as y does (same for type II pd--cycles).
*
We then define xd* and y on TdU =Ad -A as follows:

On H= En-d x(-Ad)x d, let zd be the permutation on

Qd of order p (see lemma 4.1). Define xd* and yd* the

same on 1d by sending vector En-d-lx[a,wd] to En-d-1x

[a,w zd] for all acn-Ad; then it is easily seen that

both xd* and yd* act the same as products of (p-6d) type

II p -cycles.

For each ceAd+l-{fl let W=En-d-2X{c}E n-d-1 and let

A={w}xd- i n-2 which are defined the same as in lemma 4.1.

Then by this lemma, there exists zeS n-2 which is a pd-l

cycle on A (note thatlAl=pd-l), and choose a vector vEA,

let r=pd- as in lemma 4.2. Note also that the vector v,

the element z and the set A all depend on the number

cE Ad+-{1}.

(i) If cEBdX(BdY. We defined both x* and y* on En-d-2

xcxQd+1 = AxxQ as 72 defined in (10.2). Hence, both are

products of p type II p -cycles.

(ii) If ceBdx-BdY then define x* to be 2 as in (10.2)

and define y* to be z1, in (10.1) on the set Ax2xQ.

(iii) If ceBdY-BdX then define x* to be zl, in (10.1)

and y* to be -2 in (10.2).

(iv) If ceAd+-l1-Bdx-BdY define both x* and y* as 1

(in (10.1).

Since different c gives different i.'s i=1,2,...

on different (also disjoint) sets A's. As c ranges over

the set Ad+l-{l), the union of all A's equals 8d (which is

defined in (10.18)). We have:





55

(a) xd* contains as many type I/type II p -cycles as
d
x does. yd* contains as many type I/type II p -cycles

as y does.

(b) For each cEAd+l -{}, on the set En d-2 x{c}x +i =
d+L n-d-2 l d+1
Axgxs either xd* and yd* are the same (case (i)'(iv) above)

or either one of xd* or yd* acts as zi while the other

one acts as z2 defined in (10.1) and (10.2). For c of

the latter case let gd be defined as in lemma 4.2 part (2)

on the set En-d- x{c}x d+l = Axnxn Then by part 2(b)(c)
r r
gC EN(< t vi i t *v*> ) or gC factors into product of
i=l i=l v d
r elements each lies in N() for some i. Let
c x Y
Sbe identity map if c belongs to BdnB or A d1{}-

BdX-Bdy = Cd CdY. Hence,

H gSd
(Xd) cEAd+l-{l} Yd

By induction xd* and yd* are defined;hence, let
n-1 n-1
x*= n xd* Y*= l Yd*;then x is conjugate to x* and
d=l d d=l
y is conjugate to y* also by proposition 3.4. x and x*

are conjugate, y and y* are conjugate. Hence, conjugating

elementsgl,g2,g3 can be found such that x91=x* with

gl N(T) (y*)93=y with g3EN(T) and (x*)92 = Y* where

g2 factors into product of elements each either comes
from members of F1 or members of F2. In fact, g2= II cg

The proof of theorem 2 is then completed.



11. Fusion in Direct Products

Since the Sylow p-subgroups of the symmetric groups of

degree m, where m is any positive integer, are the direct





56

products of the groups Spn's for various n's as discussed

in the preceding sections, to know the fusion of elements

in such sylow subgroups, it suffices to know the fusion

of elements in the direct product of two S n's. In this

section we will discuss the local fusion of the direct

product of two sylow subgroups Spn's with the same degree

pn. For discussion of direct product of two Spn's with

different n's, the proof is similar, hence, is omitted.

Before going into direct product, let's look at a

single sylow group Spn for a moment. Recall each con-

jugacy class of Spn, except the class containing long

pn-cycles, corresponds to a cycle decomposition, namely,

a n-tuple of constants (non-negative integers) {CC1,C2,C2...

,Cn-l} satisfying the relation

CO + PC1 + p2C2 + ... + Pn-1Cn- = n

where for each d=0,1,2,...,n-l. Cd is the number of

pd-cycles in the cycle decomposition. Note that CO+pC +

p2C2++P d = 6 d+ *d+l as defined in (10.10) which is
d+l
the number of vectors in cycles of lengths less than p

Again, we assume 16i.


the increasing sequence of sets A,A2,...,An as in (10.11).

Then we define the permutation x* using A. as set of
1
vectors in cycles of x* of lengths less than pi as we

did in section 10 with the choice of the element x in

the conjugacy class such that all cycles of x are type II

(except po-cycles). Thus all cycles of x* are type II

except p-cycles and x* is uniquely determined by the




57
conjugacy class. We call this unique element x* of the

conjugacy class the standard element of this conjugacy

class. The example below shows how the standard element

of a conjugacy class is found:

EXAMPLE 4.3: Let p=3, n=4 the conjugacy class cor-

responds to the constants C0=12,C ==2, C2=4, C3=1, contains
the standard element x* =

([1111)1112[(1113)([1121)([1122 ]Y[1123)([1131][1132][1133

([1211])([1212]X[1213]1[l221] [1222] [1223])61231] [1232][1233])

([1311][1321][1331] [1312][1322][1332] [1313][1323][1333)

([2111] [2121] [2131] [2112] [2122] [2132] [2113] [2123] [2133])

([2211][2221][2231] [2212][2222][2232] [2213][2223][2233)

([2311] [2321] [2331] [2312] [2322] [2332] [2313] [2323] [2333)

([3111] [3211] [3311] [3121] [3221] [3321] [3131] [3231] [3331]

[3112] [3212] [3312] [3122] [3222] [3322] [3132] [3232] [3332]

[3113][3213] [3313] [3123] [32231. [3323] [3133] [3233] [3333])

Notice that this example goes beyond the condition

1<6. - 1
substantial and hence the standard element can be found

in general.

Now let S and S' be Sylow p-subgroups of the sym-

metric groups ypn and ypn' acting on the set on and n '

respectively. Consider the group)= S x S', it is easy

to see, is a Sylow p-subgroup of y2.pn acting on the set

nU gn' (disjoint union),and it is also the Sylow p-subgroup
of the subgroup y nx Ypn' of Y2.pn where pnx Ypn' is

considered as the subgroup which "stabilizes" both sets

n and on'; that is, the permutations which maps elements

of Qn into n elements of n into n '.






Let the notations (5.6)-(5.10), (6.1)-(6,7), (9.1)

and (9.2) be defined on S as well as on S' with the lash

"'" added to all those in S'. Consequently, we have

notations for "products", namely, we have Ai x A',

i=0,1,...n-l, in particular T x T' = the group generated

by all tv(n-l)'s and t (n-1)' s. Also, besides F1,F2,

F 1'F2 we may define

(11.1) F* = {N(TxT')}UF1UF'

(11.2) F* = {N(TxT')}UF2UF2'

Now let xx' and yy' be two conjugate elements of
S* s*
= S x S' where x,yeS and x',y'eS'. Let x*, x y*, y

be the standard elements of the conjugacy classes containing

x,x',y,y' respectively; then, by the main theorem, there

exist gl'g 9g2 2 which factor into products of elements

in the family F ,F1',F1,Fl' respectively (or F2,F2',F2,F2'

respectively) such that

x91 = x* (x') g = x
92 9
Y 2 y (y' g2
y = y* (y') =y

hence

(11.3) (xx')191 = ((xx')l = (x l.x91 g1

= (*.')1 = (x*)91 x = x*.xI

Here the qualities hold because gl centralizes x' and

gl' centralizes x4, likewise

(11.4) (yy')292' = y*y *

Thus F (and F *, too) control the local fusion from

xx' to x*x* and from yy' to y*y' Hence, the last
1* *
problem is how the element y*y and x*x' are fused together.

We claim in the following lemma that the normalizer (N(TxT'))







will give the local fusion from x*x* to y*y'* (that is

the reason why we throw N(TxT ) into both definitions

of F* and F*):
1 2
LEMMA 4.4 Let x,yeS, x',y ES', if xx',yy'E=

SxS' are conjugate and if x*,x'*,y*,y'* are standard

elements of the conjugacy classes containing x,x',y,y'

respectively, then there is gEN(TxT') such that

(x*x'*) =y*y,*.

PROOF Since we will be dealing with the four elements

x*,x'*,y*,y' through the proof, to avoid writing the

"*" all the time, we will use x,x',y,y' to denote x*,x'*

y*,y' That is, we assume x,x',y,y' are standard elements

themselves such that xx' is conjugate to yy'.

Let Cx, C ...,C C...,C
',C f' .., C x'y o 0 !1' n-11
x' '...n- } { ,C ...,Cn } be the constants

corresponding to these conjugate classes. We have, because

xx' is conjugate to yy', that for each i=0,1,2,...,n-l

(11.5) -(Cx-C)=Cx-Cy
i 1 i i
For i=0, we have CY-CX=CX'-CY; let V be the set of
O O O O O
vectors in which are fixed by exactly one of the two

elements xx' and yy', and let V' be the set of vectors
0
in n which are fixed by exactly one of the two elements

xx' and yy'. By the way the standard elements are defined,

it is not too difficult to see that I Vo = V' =

CY-CxI=IcX'-CY'I=f We call vectors in V and V'
extra" vectors of p-cycles of xx and yy'.
"extra" vectors of po-cycles of xx' and yy'.




60

They form "blocks"{v.xv} and{w'x' } for some v and w' in

n-1 (recalling all CoX,Co,C Y,C Y' are multiples of p.

Hence, f is also a multiple of p).

Now let gl be the permutation, which acts on VOUVO'

and leaves all other vectors fixed, defined by sending

blocks onto blocks,and in each pair of corresponding

blocks vxst and w'xo' it maps [v,a] -[w',a] all aen

Then, (xx')1 and yy' have the same set of fixed vectors

and obviously geN(TxT') because if g interchanges the

blocks vxQ and w'xQ' then (tv)g = tw' and (twI') = tV.

We have used gl as conjugating element to xx' to pass

from xx' to the element (xx')gl which is also a product

of standard elements and has the same fixed vectors as

fixed vectors of yy' in the sense that they are closer

"looks."

Inductively, we find elements gl,g92,...gd all in

N(TxT') such that (xx')9192,'**9d and yy' have the same

vectors in all cycles of lengths less than pd;then let

Vd be the set of vectors in Q which lie in one and only

one pd-cycle of either (xx')gl ''-gd or yy', and let

Vd' be defined likewise. Then, define gd+l to be the

permutation acting on VdUVd' leaving all other vectors

fixed by interchanging vectors in pd-cycles of (xx')gl''''gd

and yy'rthus making the elements (xx')g91'*-gd+l and yy'

have the same vectors in all cycles of lengths less than
d+l
pd. By induction, we have the elements gl,g2,...gn all

in N(TxT'). Hence, g=g1g2...gne N(TxT') such that (xx')9=

yy'. Conclude the proof of the lemma.





61

Let's look at an example which illustrates the proof

of the lemma;

EXAMPLE 4.5 Let p=3, n=3, in S27 x S2/. The standard

elements x, x', y, y' are given by the constants:



d Cd Cd' CdY CdY'


0 3 9 6 6

1 2 3 4 1

2 2 1 1 2



We first write the cycle decomposition of xx' and yy'

then underline the cycles of xx' which are contained in

yy', and find the sets V0 and Vo' hence the conjugating

element gl. Secondly, write the cycles of (xx')gl and

underline the cycles of (xx')91, which are contained

in yy'. Hence, find the sets V1 and VI' and then the

conjugating element g2. In this example, the sets V2 and

V2' are empty;hence, (xx')1l92 = yy'. The local conjuga-

tion takes two steps to go from xx' to yy'. See the

following page.










n en en en




Nl ri n e
'-







N en N n
in N en N






















N en en e
N mI N m
en N en N







N N'


eN IN




N M N N
N en:N en
N inm N e



































I _
NM
-r r^, ru n






N rI N n

r i r-< rM

N e IN en
r L


e iN N
r I I'




























N en N e
r N | m rs m
N N N





n N N



































en N en N
N m o N































m CM H m M
i-l m Ii-4 rs



































04 04 04 rM
en Nen N
































V- i" r- 04-'
en( eNen N
m r( P i m i-































'en-
e nIN en


N enHI | en
en N; N








r -' m -< 24
N N ins~












ni~N en( N
.1N en en









en N en en
-i | 04 i-l
-i i m N rl
iit i- r 1 -







rsc~, 04040


r-i|0 -( 04
r-4 u r 4 m


i- I04 r -^

r1 .c m -
i- -
x- 04 r-


N



m N
N mn



M m







NM m






N eN
N eN


en -





























-? e
N m







N N



















C m
N rN






























en N


















e4 N
O 0

N I-
r^ i

i- r


















































6 N
N en








































en e : -





N N N












en N
-
en e


N IN
























eni I










0'1
^-l m

O ^-
m r-i
r-

r^
n rl"





m c~

r-
^ ^_ ^_

f i C
N I-(








1- m c


r-<+ ml




rc t sim



r( rti r


N'-f ~ ryl

r-i m

IN'*s..
rs V* *m m r

i- r-O r\ m






.- -i r- rM




II II
0 -0r^ r
> > I


n N






0N m


















3 0/
0N m






















N en
en N
Ni m





N 0e
Ne m









NeN
en N












N N

























en en

en en
N e
n N




























e Nn N
04 04
N m

e N
m: 1?1
















--


























N
X4m

















2
N N
m r?

m mi

04 m

nm 0





r- i N
m r-









- N













N N N en
N N















N N en N
- en NN
m ~m
rri r^











en,--.en en4N















N N en en-
e N~






m '- H-* en N -










-( N
r c-N csi 4 f


Ntr-, .-. in -



























NNi
0 0' -
m04' r


















x
rl r-< *'l" -

04 0 0 -

^ ^ -S^
(M 0
CM,, 04 p-
r- *- rri 0
(M ^ ?& ^^
04 N- +0 4 r

Fs 0 0 0


rl^ ^5. *-
CM, I-* M ft n
(M r) 0 ~
i- -^ / -
f- .5.'^-
ir **11 m t

04 oi r-> r
x tt 0





63

As a corollary of the main theorem and this lemma,

we have:

CONCLUSION: In the Sylow p-subgroups of a symmetric

group of any degree, the local fusion of elements is

given by the normalizers of groups of only two different

structures; one is a big elementary abelian group of p-power

rank, the other is the small elementary abelian group of

order p.













APPENDIX


In this appendix we will see four examples of permu-

tations x, x*, y, y* in Spn with p=5, n=3 and x, x*, y, y*

are all conjugate in Ypn such that x is conjugate to

x*. These examples illustrate: (1) how the conjugating

element glEN(T) is found to make xg1=x*; (2) how the

element g2,EN(T) is found to make (y*)93=y; (3) how the

element g2 is found so that (x*)92=y* and g2 factors into

product of elements in the normalizers of the family of

p-subgroups of the form with v(n-2)

ranges overthe set _n-2 mentioned in the main theorem.

In fact, how the elements x* and y* can be found

from x,y as mentioned in Theorem 2 can also be seen im-

plicitly from these examples.

Each example gives the element first by the maps

x0,xl,x2,...Xn_1 as defined in (5.9), then by its cycle

decomposition. All cycle decompositions are written in

the same manner such that the corresponding cycles all

have the same length and/or the same type. The type

of each cycle (or cycles) is put on the top of the cycle

(or cycles). The elements gi are defined to map the vectors

into the vectors in the corresponding positions of the other

element. The illustration of how gleN(T) and how g2 factors






65


into product of elements in N()

can be seen to some extent.






















































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REFERENCES


[1] Alperin, J. L., Sylow Intersections and Fusion,
J. Alg. 6 (1967) 222-241.

[2] Alperin, J. L., Up and Down Fusion, J. Alg. 28
(1974) 206-209.

[3] Alperin, J. L., Finite Groups Viewed Locally,
Bul. AMS 83 No.6 (1977). 1271-1285.

[4] Alperin, J. L. and Gorenstein, D., Transfer and
Fusion in Finite Groups, J. Alg. 6 (1967) 242-255.

[5] Dolan, S. W.,Local Conjugation in Finite Groups,
J. Alg. 43 (1976) 506-516.

[6] Finkel, D.,Local Control and Factorization of
Focal Subgroups, Pacific J. Math. 45 No.l
(1973) 113-128.

[7] Glauberman, G., Global and Local Properties of
Finite Groups, Academic Press, New York, 1971.

[8] Goldschmidt, D. M., A Conjugation Family for Finite
Groups, J. Alg.16 (1970) 138-142.

[9] Gorenstein, D., Finite Groups, Harper & Row,
New York, 1968.

[10] Hall, M., The Theory of Groups, Macmillan, New York,
1959.

[11] Herstein, I. N., Topics in Algebra, John Wiley & Son
Inc.,New York, 1975.

[12] Higman, D. G., Focal Series in Finite Groups,
Canadian J. Math. 5 (1953) 477-497.

[13] Huppert, B., Endliche Gruppen I, Springer-Verlag,
New York, 1967.

[14] Kaloujnine, L.,La Structure des p-groupes de Sylow
des Groupes Symetriques Fins, Ann. Sci. Ecole
Norm, Supor 65 (1948) 239-276.









[15] Kaloujnine, L., Sur les p-groupes de Sylow du Groupe
Symetriques du Degree' m, C. R. Acad. Sci. Paris
221 (1945) 222-224. P

[16] Kantor, W. M. and Seitz, G. M., Step-by-Step Conjugation
of p-subgroups of a Group, J. Alg. 16 (1970)
298-310.

[17] Passman, D., Permutation Groups, Benjamin, New York,
1968.

[18] Rotman, J., The Theory of Groups; An Introduction,
Allyn & Bacon, Boston 1968.

[19] Schreider, 0., Uber die Erweiterung von Gruppen I,
Monatsh Math. Phys. 34 (1926) 165-180.

[20] Thompson, J. G., Normal p-Complements for Finite
Groups, J. Alg. 1 (1964) 43-46.

[21] Weir, A. J., The Sylow Subgroups of the Symmetric
Groups, Proc. AMS 6 (1955) 534-541.












BIOGRAPHICAL SKETCH


John-tien Hsieh was born 1946 at Tainan City of Taiwan,

Republic of China. He studied mathematics at National

Taiwan Normal University in Taipei during the years

1965-1970 and got his B.S. degree in 1970. He came to

University of Florida as a graduate teaching assistant

in the fall of 1973 to study mathematics and got his

M.S. degree in the summer of 1974.

Mr. Hsieh is married to Lienzu Lin. They have a

daughter named Ellen who is now 6 years old. Mrs. Hsieh

is also a mathematics major graduate student studying

at the University of Florida.











I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




David A. Drake Chairman
Associate Professor of
Mathematics



I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Mark P. Hale Jr. Cochairman
Associate Professor of
Mathematics



I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality
as a dissertation for the degree of Doctor of Philosophy.




James K. Brooks (,
Professor of Mathematics











I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.




Thomas E. Bullock
Professor of Electric
Engineering



I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality
as a dissertation for the degree of Doctor of Philosophy.




Katherine B. Farmer
Assistant Professor of
Mathematics



This dissertation was submitted to the Graduate Faculty
of the Department of Mathematics in the College of
Liberal Arts and Science and to the Graduate Council,
and was accepted as partial fulfillment of the requirements
for the degree of Doctor of Philosophy.

December 1978


Dean, Graduate School






























































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