CHAPTER HI

IAN~.r~n GRAPH AUTO-MORPHISMS, CONE GRAPHS, AND I'-GROUI~

Two resul t s ma rk the origin of r e c e n t group- theore t ic approaches to g raph iso-

morphism: The labelled graph au tomorph ism problem, and the i somorphism problem

for cone graphs. This is not to say t ha t here group- theore t ic techniques are applied

to graph i somorphism for the first t ime. Rather, it appears tha t the impac t of these

resul ts has been to convince r e sea rche r s £hat a group- theore t ic approach to graph

i somorphism is a reasonable and p rac t i ca l line of a t tack. I t is also t rue t h a t cone

graphs possess a topological s t r uc tu r e which pe rmi t s visualizing a b s t r a c t p roper t ies

of the au tomorph ism group of graphs of bounded valence, and especial ly of t r iva lent

graphs.

In this chapter , we will develop both the polynomial t ime solution of the labelled

graph au tomorph i sm problem and resul t s concerning the complexi ty of an isomor-

phism t e s t for regular cone graphs. We will also discuss the relat ionship between

cone graphs and Sylow p-subgroups of pe rmu ta t i on groups, which seems to have t r ig-

gered the i somorphism tes t s for graphs of fixed valence (Chapters IV and V). Finally,

we will develop a number of basic a lgori thms for the class of p-groups. In Chapter IV

we will develop fur ther computa t iona l techniques for p-groups.

L The Labelled Graph Automorphism Problem

When designing an i somorphism t e s t for graphs, it is a na tura l idea to a t t e m p t a

ver tex classification with the aim of reducing, to a manageable magnitude, the

number of a p r ~ o ~ possible isomorphisms. That is, if X and X' are graphs to be t e s t ed

for isomorphism, we wish to partition the vertices of X and of X' into classes such that

an isomorphism can only map ~ vertex v of X into a vertex w of X' if v and w are in the

same class,

As a s imple example i l lustrat ing this idea, consider classifying ver t ices by

v~le~ce, i.e. by the number of edges incident to the ver tex classified. Obviously, no

i somorphism can map a ver tex of valence k into a ver tex of valence j~k, thus we have

61

here a sound classif icat ion cr i te r ion . Of course, if X and X' are regular graphs, i.e., if

X and X' are graphs in which every ver tex has the same valence, t h e n this ve r tex

classif icat ion yields no informat ion.

Over the two or so decades during which ver tex classif icat ion has b e e n the dom-

i nan t style of approaching graph isomorphism, m a n y e labora te c r i t e r i a for classifying

ver t ices have been proposed. Unfor tuna te ly , none of the proposed c r i t e r i a has so far

succeeded in solving the genera l problem. Therefore i t is an in t e re s t ing ques t ion to

s tudy how "good" a ver tex classif icat ion scheme has to be in o rder to serve as basis

for a polynomial t ime i somorph i sm test . This mot iva tes

Pl~B,.mm 1 (Labelled Graph Automorphism)

Let X = (V,E) be a g raph with n ver t ices , and assume t h a t V has b e e n pa r t i t i oned in to

the classes C1 . . . . . C s, forming the pa r t i t i on C, such tha t I Cit <- k, where k is a con-

s t an t i n d e p e n d e n t of n. Find all au tomorph i sms of X which setwise stabil ize the

classes C i, 1 -< i -< s. That is, find

Auto(X) = I a e Aut(X) t (Vi-<s)(VxeCi)(x a c Ci) t,

the subgroup of those a u t o m o r p h i s m s of X which r e s p e c t the pa r t i t i on 6".

In the following, we will view the par t i t ion C as the r e su l t of a vertem lab~Uing of

the g raph X with s d i s t i nc t labels, The class Ci consis ts of the ver t ices in X which

ca r ry the i TM label.

In o rder to d e m o n s t r a ~ the re la t ionship of P rob lem 1 to ver tex classif ication, we

let X be the dis joint un ion of two connec t ed graphs which are to be t e s t ed for i somor-

phism, and we fu r the r assume tha t the pa r t i t ion C is the r e su l t of a c o r r e c t ver tex

classif icat ion procedure . Then the two graphs are isomorphic iff every gene ra t ing se t

for Autc(X ) conta ins a t l eas t one p e r m u t a t i o n which exchanges the connec ted com-

ponen t s of X.

I. 1. A Determinis t ic Algorithm for Problem 1

We will show that Problem I has a deterministic polynomial time solution.

Specifically, we will apply the techniques of Chapter If, Section 4, and demonstrate

that we can make Autc(X ) (k,c)-accessible.

62

Recall the def ini t ion of (k ,c)-access ibi l i ty (Chapter II, Definition 22). We will show

t h a t hutc(X) is (2,c)-accessible (for some c o n s t a n t c) by t r app ing i t as the subgroup

G (r) in a subgroup tower

I = G (m) <: • " " < G (r) = A u t c ( X ) <: • -" < G (1) = G

of a known group G.

Intui t ively, we le t the groups G (ra), G (m-l) . . . . . G (r+i) be the pointwise s tabi l izers , in

G (r), of the ver t ices in the classes C i of X, s tabi l izing every ver tex in an individual class

at each step. Clearly we have s imple m e m b e r s h i p t e s t s for these groups.

The groups G (r-l) . . . . . G (1) are ob ta ined as the a u t o m o r p h i s m groups of ce r t a in

label led graphs Xj derived f rom X. Here, the t r ick is to define the graphs Xj such t h a t

Autc(Xj+l) is a subgroup of Autc(Xj).

In par t i cu la r , we define X1 to be the graph X s t r ipped of all edges. Note t ha t

G (1) = Auto(X1) is the d i rec t p r o d u c t of symmet r i c groups act ing on the individual ver-

tex classes of X, i.e.,

Auto(X1) = G (1) = r lSym(Ci) i = l

F u r t h e r m o r e , s ince Aute(X1) r e spec t s the label classes, Auto(X) is a subgroup of

Auto(X1). Note t ha t we have gene ra to r s for G O), and tha t we can easily t e s t m e m b e r -

ship in this group.

The graphs X2 . . . . . X r are ob t a ined by gradua l ly adding back in to X 1 the edges of X.

Here it is c ruc ia l to add edges in batches, where each ba t ch consis ts of all edges con-

nec t ing the ver t i ces m two classes, C h and C i, This ensu re s t h a t Autc(Xj+1) is a sub-

group of Autc(Xj).

We now formal ly specify the c o n s t r u c t i o n j u s t out l ined. Let X = (V,E) be the

g raph u n d e r cons ide ra t i on IVI = n, and le t the pa r t i t i on of V be C = fC 1 . . . . . Csl,

where I Cil -- k, I --- i-< s.

Define El, i = t(v,w) E E v E Ci, w E C j / , and l e t X l = (V,F1),where F1 = ¢ . Define

the graphs Xj = (V,Fj), 1 < j ~ r, r = (~)+1, by

Xu = (V,F~) = (V,F1uEI,1)

X 3 = (V,F~) = (V,F~)EI,~)

Xs+1 = (V,Fs+I) = (V, FsUEI,s)

63

Xs+~ = (V,Fs+~) = (Y,Fs+z uE~,~)

Xs+ S = (V,Fs+3) = (V, Fs+euE2,3)

Xr = (V,Fr) = (V,Fr-IUEs,,) = (¥,E) = X

Fur the rmore , we define G(J) = Autc(Xi).

~ I ~ 1

Let X = (t l . . . . . 81, E) be a g raph with E = I(1,2), (1,4), (2,3), (~,6), (2,7), (~,8), (3,5),

(3,8), (4,5), (5,6), (6,7), (6,8)], and with the label classes C 1 = fI,2,8{, C 2 = ~4,5,8{, and

Ca = t7,81. X is shown in Figure I below. Then the g raph sequence X1 .. . . . ?(7 is defined

by the edge sets

F 1 = ¢

F~ = I(1,~), (~,a)~

Fa = Fe U t(1,4), (2,6), (3,5)t

F4 = Fa u t(2,7), (~,8), (a,a)t

F5 = F4 u 1(4,5), (5,6)1

F8 = F5 U 1(6,7), (6,a)l

F T = F e u ¢ = E

Cl

I t f ' ,

'41 I I I i

i I I !

151 2*

I !

ca /

C~

The graph X

Figure i

F6 = F7 since t he re is no edge (7,8) in E. Note t h a t X7 = (V, FT) is the g r a p h X. As illus-

trat ion, the g raph )(3 = (V,F3) is shown in Figure 2 below. Observe t h a t the only edges

C1

64

1 '- ( 4 ] T i

I'! ; I I

I 3 ,-r"~ ~- - t6 ,

;7 I

i C3 i

18

The g raph ]{3

Figure 2

03

p r e s e n t in X3 are those edges (v,w) of E such t h a t v C C1 and w E C1LjCa. []

Having made these definit ions, we need to verify t h a t the resu l t ing subgroup

tower makes Autc(X) accessible , so t ha t we can apply Theorem 12 of Chapter II, and

d e t e r m i n e Autc(X) in po lynomia l t ime.

L~m~r~A 1

G (1) is the d i rec t p r o d u c t of the s y m m e t r i c groups Sym(Ci) act ing on the individual

s

label classes, i.e., G (1) = ~-[Sym(Ci). i = 1

Proof X1 has no edges, and G (0 = Autc(Xl), by definit ion. -

L ~ X A 2

For 1 ~ j < r, G (j÷l) is a subgroup of G(J) of index at mos t (k!) 2.

Proof Let Xj+ 1 = (V,FjL)Eha). Since Fj con ta ins no edge connec t i ng ve r t i ces in C h

with ver t ices in Ci, any a u t o m o r p h i s m of the label led g raph Xj+I is also an au tomor -

ph i sm of the label led graph Xj, and so G (j+O < G (j). F u r t h e r m o r e , if ~, ~ E G O) such

tha t ~r3~ -I s tabi l izes the ver t ices of ChuC i pointwise, t h e n ~ and ~ m u s t be in the same

r igh t coset of G 0+l) in G 0). Since t he r e are a t mos t (k!) ~ d i s t i nc t ways of p e r m u t i n g

the ver t ices in C h and in C i, the index (G(J):G U+1)) c a n n o t exceed this bound. "

We now t u r n to the lower p a r t of the subgroup tower. For 1 -- j -- s, we define

c (r+j) = { ~ ~ G (r+j-~) i ( V x e q ) ( x " = x)

65

i That is, t h e group G (r+9 is the pointwise stabil izer in Auto(X) of the se t uCi .

~ffil

Recall t ha t r = (~)+1, i.e., r is O(n~). Therefore, we have jus t defined a subgroup

tower of h e i g h t m = and so m is i.e. p o l y n o m i a l in n.

L ~ M A 3

For 1 ~ j ~ s, the index of G (r+D in G (r+j-l) is at most k!.

Proof I f #, 3~ EG (r+j-1) such t ha t ~ - 1 stabi l izes every ve r t ex in C], t hen ~ arid

lie in the same r ight cose t of G (r÷j), Since i Cj I ~ k, the bound follows. -

At this point, we have establ ished the following: Autc(X ) = G (r) occurs in a sub-

group tower of a group G = G O) of degree n for which we have a genera t ing se t of 2s

pe rmuta t ions . Note t h a t s is O(n). The height of this tower is less t han n 2, and its

width is a t m o s t (k!) 2, Since we assume tha t k is a constant , the tower is of cons t an t

width and polynomial height.

It remains to establ ish a bound for tes t ing m e m b e r s h i p in the groups G (i). Essen-

tially, we tes t m e m b e r s h i p by applying the pe rmu ta t i on to the g raph X followed by

verifying which edges have been preserved. Here we can take advantage of the fac t

that we tes t m e m b e r s h i p in G (j+l) only for pe rmuta t ions in G (D. If Xj+I = (V,FjuEk~),

t hen we need to verify t h a t up to k ~ edges, connect ing ver t ices in C h with ver t ices in

Ci, have been p rese rved by the pe rmu ta t i on in G(J). Thus, m e m b e r s h i p can be t e s t ed

inc rementa l ly in O(k 2) steps. Similarly, if we t e s t m e m b e r s h i p in G (r+j), we only have

to verify t h a t every ver tex of Cj remains fixed.

Recall Definition 22 of Chapter II: We have just shown t h a t Autc(X ) is (2,e.(k0Z) -

aecessible, where c is a cons tan t independen t of n and k. Consequently, by Theorem

12 of Chapter II, P rob lem 1 has a polynomial t ime solution. Applying Proposi t ion 5

r a t h e r than Theorem 12 of Chapter II, we obtain a sha rpe r es t imate of the t ime

required to solve Prob lem 1:

THEOR~ 1 (Bahai, Furst, Hoperoft, Luks)

Let X = (V,E) be a g raph with n vert ices , and assume t h a t V has been par t i t ioned into

the classes C = tCI ..... Cst, where, for 1 -< i ~ s, t Cil -< k. Then gene ra to r s for Autc(X)

can be d e t e r m i n e d in O(nS.(k!)8.(n+ke)) steps.

Proof Observe first t ha t G is of degree n, thus the group opera t ion requires O(n)

steps, as does comput ing t h e inverse of a pe rm u ta t i on in G. We use Algori thm 7 of

Chapter II, t rapping Autc(X ) in the subgroup tower of G defined above. Then Autc(X )

66

is g e n e r a t e d by ~ Uj~ where Uj is a c o m p l e t e r i g h t t r a n s v e r s a l for G 0+i) in GO). By j = r + s - t

L e m m a t a 1, 2, and 8, and by P ropos i t i on 5 of Chap t e r II, the t ime bound follows with

[K] = 0(n), m = 0(nZ), w = (k!) ~, t = 0(n), and T = O(kZ), =

1.2. A Random Algorithm

We will now d e s c r i b e a r a n d o m po lynomia l t ime a l g o r i t h m for P r o b l e m i. Such an

a l g o r i t h m is of i n t e r e s t for two r easons : F i r s t , i t has an e x p e c t e d runn ing t i m e m u c h

b e t t e r t h a n t h e d e t e r m i n i s t i c vers ion , and secondly , i t gives us t he o p p o r t u n i t y to

show how to g e n e r a t e r a n d o m e l e m e n t s of a p e r m u t a t i o n group with a un i fo rm d i s t r i -

but ion .

Throughou t th is sec t ion , e = 2.718281828,.. d e n o t e s Eulerb constant, and In(x)

d e n o t e s the natural logariAhrn of x, i.e., t he l o g a r i t h m base e,

Let X = (V,E) be a g r a p h with v e r t e x p a r t i t i o n C = tC I . . . . . Csl, where t he v e r t e x

c lasses C i a re un i fo rmly bounded in size by the c o n s t a n t k. Recal l the def ini t ion of

the s u b g r o u p tower

I = G (ra)< - . . < G ( r ) = A u t c ( X ) < ' ' - < G ( I )= G

t r a p p i n g Auto(X) as G (r), r = (~)+1. The r a n d o m a l g o r i t h m to be d e s c r i b e d a t t e m p t s

to d e t e r m i n e t h e above s u b g r o u p tower . I t has two o u t c o m e s : e i the r , t he a l g o r i t h m

c o r r e c t l y d e t e r m i n e s a t ab l e M con ta in ing c o m p l e t e r i g h t t r a n s v e r s a l s for t he sub-

g roup tower , t h e r e b y finding a g e n e r a t i n g se t for Autc(X ), or i t c o r r e c t l y r ecogn ize s

t h a t the t ab le M is as y e t i ncomple t e . In the l a t t e r case , the a l g o r i t h m con t inues run-

ning. Thus, we will d e s c r i b e a r a n d o m a lgo r i t hm which always d e t e r m i n e s g e n e r a t o r s

for Aute(X ) c o r r e c t l y , b u t only with an e x p e c t e d po lynomia l r unn ing t ime . We will now

out l ine t he des ign of th is a lgo r i thm.

Intui t ively , the a l g o r i t h m d e t e r m i n e s the t ab le M for the subg roup tower of G by

sif t ing r a n d o m l y g e n e r a t e d e l e m e n t s of G, un t i l M is comple t e . Ra the r t h a n ver i fying

the c o m p l e t e n e s s of M with t he he lp of Theo rem 11 of C h a p t e r II, i.e. by pa i r p r o d u c t rfl--1

fo rma t ion , we t e s t w h e t h e r l~I is c o m p l e t e by c o m p a r i n g the p r o d u c t ~ ni wi th t h e i= 1

o r d e r of G. Here , n i is the l eng th of row i in the t ab le M. Clearly, M cons i s t s of com-

r n - I s

ple te r igh t t r a n s v e r s a l s iff IGt = ~ ni. Note t h a t [GI = 1-I(ki!), where ICif = ki, s ince i=l i=l

67

G = ~ISym(Ci). Thus the order of G is known beforehand. i= l

For a comple te specif icat ion and analysis of the a lgori thm, we need to

(a) show how to gene ra t e uni formly d i s t r ibu ted r a n d o m e l emen t s of G,

(b) prove t ha t sifting these r a n d o m e lemen t s resu l t s in un i formly d i s t r ibu ted

r a n d o m e l emen t s in each of the groups G0) in the sifting process , and

(c) give an e s t ima te of the probabi l i ty t ha t M is comple te as a func t ion of the

n u m b e r of the e l emen t s of G sifted.

We begin by showing how to gene ra t e un i fo rmly d i s t r ibu ted r a n d o m e l e m e n t s of G.

Since G is the d i rec t p roduc t of symmet r i c groups, i t suffices to show how to genera te

r a n d o m e l emen t s in S n with un i form dis t r ibut ion . However, the me thod we are abou t

to give for this t ask can Mso be applied to gene ra t e un i formly d i s t r ibu ted r a n d o m ele-

m e n t s in every p e r m u t a t i o n group for which gene ra to r s are known.

I . ~ I A 4

For fixed n -> 2, we can gene ra t e un i formly d i s t r ibu ted r a n d o m in tegers in the in te rva l

[1 . . . . . n] with an expected n u m b e r of 2t coin tosses per gene ra t ed r a n d o m n u m b e r ,

where t = [ l ogz (n - l ) ]+ I.

Proof With a b a t c h of t coin tosses, we can gene ra t e with un i fo rm d i s t r ibu t ion a

r a n d o m in teger in the in te rva l [0 . . . . . 2 t - l ] . We gene ra t e in this way the r a n d o m

in tegers r l . . . . . r I where r i is the first in teger in the sequence which is smal le r t h a n n.

We then ou tpu t r i+ l , which is in the in terva l [1 . . . . . n].

If n is a power of 8, t hen r l is always less t h a n n. Otherwise, since n - l > 2 t - l , the

probabi l i ty of r e t u r n i n g - r i exceeds 1 - 1 . . Thus the expec ted n u m b e r of coin tosses is 2~

5t --,- = St. " i=O 8 ~

THEOREM 2 (Hoffmann)

We can gene ra t e un i formly d i s t r ibu ted r a n d o m p e r m u t a t i o n s in S n with an expec ted

n u m b e r of 0(n.loge(n)) s teps and an expected n u m b e r of less t h a n 2n.log2(Sn) coin

tosses pe r g e n e r a t e d p e r m u t a t i o n .

Proof Using Lemma 4, we gene ra t e the r a n d o m in tegers ¢1 . . . . . rn-~, where rj is

in the in terva l [j . . . . . n]. We t h e n ou tpu t the p e r m u t a t i o n

vr = (n-l,rn_l)(n-8,rn_2) . - • (l,rl).

For the t iming, observe first t h a t the gene ra t ion of the n u m b e r s rj r equ i res

68

n-2 , 1 = ~. ~ ([ log~(n-D + i) coin tosses, and O(to) steps. Now j--1

< 2/log2(x)dx + 2n 1

= 2n.logz(n ) -- ~n.log2(e ) + 2n

< 2n.loge(n ) + 2n

Since c o n s t r u c t i n g ~ f rom the r a n d o m n u m b e r rj r equ i res an addi t iona l O(n) s teps,

the bounds follow.

To see t h a t the p e r m u t a t i o n s ~ so c o n s t r u c t e d are un i formly d i s t r i bu t ed ele-

m e n t s of S n, cons ide r the following subgroup tower of S n

I = G (n+l) = G (n) < " " " < G (z)< G (I) = Sn

where G (j)= Sym(Ij . . . . . nt). Note t h a t (G(D:G (j+O) = n - j + l , and t h a t the sets

Uj = t (j,i) i i c [j . . . . . n i l a re comple te r igh t t r ansve r sa l s for G 0+~) in G (D. Therefore,

every p e r m u t a t i o n , ~ S n is the un ique p r o d u c t @ n ~ n - l " ' ' @I where @j c Uj.

Observing t h a t @n = 0 , we see t h a t the lr g e n e r a t e d above are un i f o r ml y d i s t r i bu t e d

because the rj a re un i formly d i s t r i b u t e d in [j . . . . . n], and the cosets of a subgroup are

all of equal cardinal i ty . ®

CORO~ l

Let C = [C 1 . . . . . Cs~ be a pa r t i t i on of a se t V of size n, where ICi! ~ k , and le t G =

8

~Sym(Ci ) . Then we can gene ra t e un i formly d i s t r i bu t ed r a n d o m e l e m e n t s in G with i=1

an expec ted n u m b e r of n4og~(2k) coin tosses, and O(n.log~(k)) c o m p u t a t i o n steps.

Proof If ]Ci] = 1, t h e n Sym(Ci) = I, so we need only gene ra t e p e r m u t a t i o n s in a t

n mos t ~ - groups Sym(Ci). Thus, the b o u n d follows f rom Theorem 2. "

The m e t h o d of Theorem 2 is easily genera l i zed to a r b i t r a r y p e r m u t a t i o n groups

with known genera tors : Using Algori thm 3 of Chapter It, we first find a r e p r e s e n t a t i o n

ma t r ix M for G. If row i of M has n i > 1 n o n e m p t y en t r i es , we gene ra t e a r a n d o m

n u m b e r ri in the in te rva l [1 . . . . . ni]. The n u m b e r ri specifies which n o n e m p t y e n t r y ~i

in ~I ( e n u m e r a t i n g these en t r i e s f rom left to r ight) is to be se lec ted for c o n s t r u c t i n g

= 3 ~ n ' ' ' ~ 1 - Note t h a t for t r ivial rows in M we always se lec t 3P i = 0- Since

ni-< n - i + l , the bound of Theorem 1 suffices as e s t ima te for the expec ted n u m b e r of

69

coin tosses for each e l e m e n t ~ of G so gene ra t ed .

Having shown how to gene ra t e un i formly d i s t r i bu t ed r a n d o m e l e m e n t s of a per -

m u t a t i o n group G = <K>, we now es tab l i sh t h a t we can use this m e c h a n i s m to gen-

e ra te un i fo rmly d i s t r i bu t ed r a n d o m e l emen t s of a subgroup H of G, provided we have

a eompte te r ight t r ansve r sa l for H in G and can t e s t m e m b e r s h i p in H. Note, t ha t the

me thod for this task is essent ia l ly a stage of the sifting p rocedure (Algorithm 6 of

Chapter II).

Lzuea 5 (Sabai)

Let G be a permutation group of degree n, H a subgroup of index w in G, and U a com-

plete r ight t r ansve r sa l for H in G. If R is a set of un i fo rmly d i s t r i bu t e d r a n d o m ele-

m e n t s of G, t h e n we can find a set R' of un i formly d i s t r ibu ted r a n d o m e l e m e n t s of H in

O(IRI-w-(T+n)) steps, where T is the n u m b e r of s teps requ i red to Lest m e m b e r s h i p in

H.

Proof We let R'=~Tr3b~ -1 I ~rER,~P~EU, lr3b~ - I ~ H I . It is c lear t ha t R' can be

c o n s t r u c t e d in the s t a t ed t ime bound. Since the e l emen t s in R are un i formly dis t r i -

bu ted in G and since the cosets of H are all of un i fo rm size, it follows tha t R' consis ts

of un i formly d i s t r ibu ted r a n d o m e l emen t s of H. •

Recall t ha t the a lgor i thm to be descr ibed sifts r a n d o m e l emen t s of the group G in

an a t t e m p t to d e t e r m i n e a comple te table M of cose t r e p r e s e n t a t i v e s for the sub-

group tower

I = G (m) < • - - < G (I) = G

We now t u r n to e s t ima t ing the probabi l i ty t ha t M is comple te , as a f unc t i on of the

n u m b e r of r a n d o m e l emen t s sifted.

l . z u u A 6 (Babai)

Let G be a permutation group of degree n, H a subgroup of G of index not exceeding w.

Then a set R of uniformly distributed random elements of G contains a complete right

transversal for H in G with probability exceeding 1-e -q, provided that R does not con-

tain fewer than w.(In(w)+q) elements.

Proof Since the elements of R have a uniform distribution in G, the probability

that no w E R is in the right coset H~ of H in G is

_IR] ( l _ ~ [ R t ~ - 1 ~IRi < e w

70

Thus, the probabi l i ty tha t R does not contain a complete r ight t ransversal for H in G is

less than w.e w s

We will apply Lemmata 5 and 6 and estimate the probability of deriving a table M

containing complete right transversals for a subgroup tower of G, where M is obtained

by sifting random elements in G.

THEOREM 3 (Bahai)

Let G be a permutation group with the subgroup tower

I = G (m) < ' ' " < G (I) = G

where (G(i):G (i+~)) -< w. If we sift a set R of uniformly distributed random elements of

G, of size I RI -> w'(in(w)+in(p'm)), then the resulting table M contains complete right

transversals for G 0+I) in G (i), i - i < m, with probability exceeding i -i-- P

Proof By Lemma 5, we consider, at each stage of the sifting process, uniformly

distributed random elements of G (i), By Lemma 6, therefore, the probability of

obtaining an incomplete right transversal for G (i+I) in G 0) is less than I Thus, the p.m

probability that M is incomplete is less than i__ . P

We now specify the probabilistic algorithm for Problem I.

71

ALC~IaTI~ I ( P r o b a b i l i s t i c Method fo r P r o b l e m 1)

I n p u t Graph X = (V,E) with n ve r t i ces , and the v e r t e x p a r t i t i o n C = tC 1 . . . . . C,{,

where 1Cil -< k.

O u t p u t A t ab le M conta in ing c o m p l e t e r i gh t t r a n s v e r s a l s for t he s u b g r o u p tower

I= G (m) < "-' < G 0) = G defined above, where G (r)=Autc(X),r= (~)+i.

Comment Algorithm 6 of Chapter II is used as subroutine.

Method

1. begin

2. In i t ia l ize M to con t a in the i d e n t i t y p e r m u t a t i o n in row i, 1 -< i < m; S

3. Compu te N = ~ ( k i ! ) , where ICil = kf; i = l

c o m m e n t N is the o r d e r of G; m-- ]

4. whi le ( ~I ni) < N, where n i is the l eng th of row i in M, do b e g i n i = ]

5. G e n e r a t e a se t R of un i fo rmly d i s t r i b u t e d r a n d o m e l e m e n t s of G, of size

I RI = [k!.(ln(k!) +ln(2m))];

6. Sift R using Algor i thm 6 of C h a p t e r II;

7. end;

8. output(M);

9. end.

The correctness of Algorithm i is elementary. Summarizing the results established

above, we obtain the following analysis of Algorithm I:

TflEomm 4

Algorithm l uses an expected number of O(n.k!.k.logz(n.k).log2(k)) coin tosses, and

has an expected running time of 0(nZ.(k!)S.k-(n+k2).logz(n.k)).

Proof We first estimate the expected time required by Algorithm I. Let T! be

the required time to execute the while-loop once. By Theorem 8, since

(G(i):G (i+l)) ~ k!, the probability of having to repeat the while-loop is less than 1 2--n so

the expected time spent in the while loop is T z = " T~ = 2"TI. T1 is now determined

as follows:

72

m - - i

O(m-(k!)~J s teps suffice to compu te 1~ hi. i = l

O(k!.(in(k!)+in(2m)).n.loga(k)) steps suffice to generate R, by Corollary I.

O(k!.(In(k!)+In(2m)).m.(k!)~.(n+k~)) steps suffice to sift R, observing that group

operations in G require O(n) steps, and membership of elements of G (i) in G (i+l)

can be tested in O(k 2) steps.

Note that sifting is asymptotically the dominant step. Recalling that m is O(n2), T I is

therefore O(nZ.(k!)3-k.(n+k2).log~(n-k)). Clearly, the time T I dominates the time

required for all steps outside the loop, and since T 2 = 2T I, the time bound follows.

Next, it is clear that we have an expected number of 2.k!.(In(k!)+in(2mJ) permu-

tations to sift. By Corollary 2, this requires an expected number of

~.k!.(In(k!)+ln(2m)+ l).n-logz(k) coin tosses, so that we require an expected number of

O(n.kbk.ln(k.mJqog2(k)) of coin tosses. "

Theorem 4 should be compared to Theorem I. We see here that the probabilistic

version for determining Autc(X) is far superior to the deterministic version. This

means, that the probabilistie version has practical significance, and is most likely the

preferred method to be implemented.

2. Cone Graphs and Regular Cone Graphs

The tree isomorphism algorithm is one of the oldest efficient isomorphism tests

for a special class of graphs. Since isomorphism of trees can be tested so efficiently,

whereas isomorphism of graphs in general seems very much harder, it is interesting

to ask which topological p roper t i e s in t r ees make i somorph i sm tes t ing easier.

While we have no definitive answer to this quest ion, it appears t ha t the un iqueness

of sho r t e s t pa ths in t r ees is a s t r u c t u r a l cha rac te r i s t i c which somehow helps.

Accordingly, we will make a p r e l imina ry analysis of a b roader class of graphs,

cons is t ing of c o n n e c t e d g raphs in which the re exists a ve r tex v0, such tha t , for every

ver tex w in the graph, t he re is a un ique s h o r t e s t pa th be tween v o and w. We call such

graphs cone graphs, and we cal l the ve r t ex v 0 a root of the graph.

The g roup- theore t i c m a c h i n e r y developed thus far t u r n s ou t to be insuff ic ient to

handle cone graphs in general , and we will the re fore r e t u r n to the s tudy of this class

in Chapters W and V with more advanced techniques . Here we will prove some resu l t s

73

about the s t r u c t u r e of the a u t o m o r p h i s m group of cone g raphs wi thout developing a

specific a lgor i thm. A specia l case of some i m p o r t a n c e to t h e m a t e r i a l of Sec t ion 3

below is the s t r u c t u r e of the a u t o m o r p h i s m group of reg-u/~zr cone graphs: Let X be a

cone g r a p h with roo t Vo, and cons ide r the sub t r ee of X consis t ing of all s h o r t e s t pa ths

f rom v 0 to eve ry v e r t e x of the graph. Then X is a r egu la r cone g r aph if the v e r t i c e s in

the t r e e which are at the same d is tance f rom v 0 have an equal n u m b e r of sons. Note

t h a t r egu la r i t y does not imply t h a t such v e r t i c e s are i nc iden t to an equal n u m b e r of

edges, s ince we a re not c o n c e r n e d about non t r ee edges in X.

Let X = (V,E) be a graph, u and w ve r t i ce s i n X . Apa~/~ be tw een u and w is a

s equence v0, v I . . . . . v k of ve r t i c e s of X such t h a t v 0 = u, v k = w, and (v~_i,vi) is an edge

of X, I -< i -~ k. The l e~g th of t he pa th is k. F u r t h e r m o r e , if all the ve r t i c e s vi a re dis-

t inct , t h e n the pa th is s~rnple.

A pa th v0 . . . . , vk in X is a shortest pa th if t h e r e is no pa th b e t w e e n vo and vk of

length less t han k. In this case, k is the d~stu~ce of v o f r o m v k. Note t h a t in genera l

s h o r t e s t pa ths are n o t unique. However, if vo . . . . . vk is a s h o r t e s t p a t h be tween Vo and

vl~, t h e n v 0 . . . . . v i is a sho r t e s t pa th be tween v o and v i, i < k. Therefore , if X is a con-

n e c t e d g raph and v 0 is a fixed v e r t e x in X, t hen i t is always possible to s e l ec t a se t of

s h o r t e s t pa ths be tween Vo and eve ry o the r v e r t e x in X such t h a t the e d g e s of t h e s e

s h o r t e s t pa ths span a t r e e in X. This t r e e is cal led a ~readt/z-jlrst-sesre/~ t ree f r o m v 0,

h e r e a f t e r abb rev i a t ed BFS-tree . There is an O([VI+]E] ) a lgo r i thm for cons t ruc t ing a

BFS- t ree of a g raph X = (V,E) f rom a g iven v e r t e x Vo..

DEHNrrIOI~ 1

A c o n n e c t e d g r aph X = (V,E) is a co~e /Traph if t he r e exists a v e r t e x v 0 ~ V, such t h a t

for every w E V the re is exac t ly one s h o r t e s t pa th be tween v 0 and w. The v e r t e x v 0 is

cal led a root of the cone graph.

F igures 3 to 6 below give examples of cone graphs. F igures 7 and 8 give example s

of graphs which are not cone graphs. Note tha t a cone g raph may have severa l roots.

For example , for the g raph in Figure 3 every v e r t e x is a root .

Dmm~TIOH 2

A cone g r aph is reg~LaT if, for at l eas t one of i ts roo t s v 0, the v e r t i c e s in the BFS- t ree

of equal d i s tance f r o m v 0 have an equal n u m b e r of sons.

For example , the cone g raphs in Figures 3 to 5 are regular , whereas the cone

g raph in Figure 6 is not. Note t h a t i t is s o m e t i m e s possible to c o n v e r t a g raph X into

a cone g raph by adding a new v e r t e x as midpo in t of an edge of X. For example , the

74

~ I 0 ,,.,/ \/ \/

The P e t e r s e n Graph

Figure 3

cone graph in Figure 4 has been obtained from Ks, s in Figure 7 by adding the vertex 7

as midpoint dividing the edge (1,8).

A p a r a m e t e r which affects the s t r u c t u r e of the a u t o m o r p h i s m group (and also the

eff ic iency of the i s o m o r p h i s m t e s t s in C h a p t e r s IV and V) is t he l a r g e s t n u m b e r of

3 4 ~ 5 6

\ / \ / i 2

\ / 7

K3,3 modi f ied

F igure 4

sons of any interior vertex in the BFS-tree.

D~I~NITION 3

The degree of the cone g r a p h X with roo t v is the l a r g e s t n u m b e r of sons any v e r t e x of

X has in the BFS-tree from v. The he/ght of X is the height of the BFS-tree.

For example , t he cone g r a p h of F igure 3 is of d e g r e e 3 and of he igh t 2, whe rea s

the cone g r a p h o~ F igure 4 is of d e g r e e ~ and of he igh t ~. A cone g r a p h of d e g r e e

75

must necessarily be a nonbranehing tree.

5 6 ~ 7 ~ f ~ ~ "",',',' • 9 ~ ~ 1 0

\7 \/ V 1

Figure 5

° °

\ / 2 3 ..... 4

Figure 6

//5 6

~3,3 Figure ?

76

2

6 7 8

I1 12

Figure 8

2.1. The Structure of the Automcrphism Croup of Cone Craphs c~ F~_xed I)e~ree

We will analyze the s t r u c t u r e of the a u t o m o r p h i s m group of cone graphs using the

t echn iques of Sec t ion 1. In pa r t i cu la r , we will cons ider the following

P R O B ~ 2

Given a cone g raph X with root v and of degree d, where d is a cons tan t , d e t e r m i n e

gene ra to r s for Autv(X), the group of all a u t o m o r p h i s m s of X which fix the root v.

It is no t ha rd to show tha t an efficient a lgor i thm for P r ob l e m 2 can be used to

design an efficient i somorph i sm tes t for regu la r cone graphs of fixed degree, as well

as to d e t e r m i n e gene ra to r s for the full a u t o m o r p h i s m group. We omit the proof of

these e l e m e n t a r y resul ts .

We will now descr ibe how to t rap sec t ions of the a u t o m o r p h i s m group in var ious

subgroup towers. We will use a col lect ion of ve r tex pa r t i t i ons to define these sub-

group towers, and discuss some of the difficulties e n c o u n t e r e d when d e t e r m i n i n g

these towers efficiently. Here we find it conven i en t to visualize cone graphs

77

geomet r ica l ly as drawn in a specific way. In par t icu la r , we draw cone graphs so t ha t

ver t ices of equal d is tance f rom the root v are l ined up horizontal ly, and the BFS-tree

is drawn as a p lanar graph growing upwards, as shown schemat ica l ly in Figure 9

below.

ver t ices of d i s tance k f rom v

A cone g raph

Figure 9

If X = (V,E) is a cone graph with root v and height h, then the vertex set V is parti-

tioned into sets Vk, 0 -< k - h, where V k consists of all vertices in V which are at dis-

tance k from the root v. Let u and w be vertices in V k, We sometimes need to set up a

l - I correspondence of vertices in the subtree rooted in u with vertices in the subtree

rooted in w, if such a correspondence exists. We do this by pairing, left to right, des-

cendants of u with descendants of w which are at the same distance from u and from

w, respectively. For example, in the cone graph of Figure 3, letting u=2 and w=3, the

corresponding pairs of descendants are (5,7) and (6,8).

DEFINITION 4

Let X = (V,E) be a cone graph with root v. A %-=~toTnorpA/sm of X (with respect to v)

is a permutation a of V such that, for all vertices u in X of distance k or less from v,

u a = u. A(k)(x) is the group of k-automorphisms of X.

Note that k-automorphism is always defined with respect to a fixed root of the

graph. When the graph X is clear from the context, we will write A (k) instead of A(k)(X).

Note t h a t A (°) = Autv(X ).

EXal~I~ 2

Let X be the cone graph of Figure 3 above. Then A (2) is the t r ivial group I, s ince its

e l emen t s m u s t fix every graph vertex. The group A (1) is ge ne r a t e d by (5,6)(7,8)(9,I0),

and is of o rder Z. The group A (°) = Autl(X) is of o rder 12, and is g e n e r a t e d by the

three p e r m u t a t i o n s (2,3)(5,7)(6,8)(9,10), (2,3,4)(5,7,9,6,8,10), and (5,6)(7,8)(9,10). [3

78

A ~÷]) is a normal subgroup of A (k).

Proof Recall that Autv(X) setwise stabilizes the vertices in Vk, 0-~ k ~ h. We

therefore identify A (k) as the setwise stabilizer of Vk+ i in A (k), and A (k+l) as the point-

wise stabilizer of Vk+ I in A (k). Thus A (k+i) 4 A (k), (cf. Chapter If, Subsection 1.4,

Definit ion t l ) . -

DEFn~rnoN 5

Let X = (V,E) and X' = (V',E') be two cone graphs with roots v and v', respectively, and

with NFS-trees of height h. Assume that X and X' contain an equal number of vertices

of equal distance from the root, and let ~ be a fixed but arbitrary I-i map from V k

onto V'k, 0 --< k -< h. Then X and X' are Ic4sornorp~%~c (with respect to ~c) if there is an

isomorphism ~ from X to X' such that, for u e Vj, j 4- k, u ~ = u ~.

Clearly X and X' are h-isomorphic if[ m is an isomorphism. In the sequel, we let Ic

be the i-i correspondence obtained by pairing vertices of equal distance in the two

graphs, from left to right, as explained above.

EXAMPLE 3

Let X and X' be the cone graphs shown in Figure i0 below, with the roots i and I'. Let

\ / \ / 2 3

4 ~ ' 7'

\ / \ /

l '

Figzu-e 10

m a p i to i', i ~ i -< 7. Then X and X' a r e 0 - i somorph ic , b u t a r e no t 1- i somorphic .

Let X -- (V,E) be a cone g r a p h of deg ree d and he igh t h wi th r e s p e c t to t he roo t v.

Recal l Defini t ion 4. The g roups A (k) f o rm the subg roup tower

I = A (h) <~A (h-i) <~ , • - <~A (0) = Autv(X ),

where A (k) is the pointwise stabilizer of all those vertices in X whose distance from the

79

root v is k or less.

It s eems imposs ib le to apply the t echn iques of Sect ion I to this subgroup tower

directly, as the re seems to be no good way to ex tend the tower to a group with k n o ~

generators or of known order. Furthermore, the index (A(k):A (k+1)) need not be small.

So, we p lan to d e t e r m i n e the groups A (k) separate ly . De te rmin ing gene ra to r s for A (k)

will requi re a r ecu r s ion which will be explained below.

Intuit ively, A (k) is t r apped using the ver tex pa r t i t i on C k, where all ve r t i ces of dis-

t ance k - 1 or less f rom the root are in separa te blocks of size 1. F u r t h e r m o r e , for

each ver tex v~ E V k, the re is a block conta in ing precise ly v i and all ver t ices in the sub-

t ree roo ted in v i. Note tha t C k is ob ta ined f rom Ck+ 1 by merging , for each ver tex

v i E V~, the block in Ck+l which con ta ins vi with the blocks conta in ing the sons of vi.

This pa r t i t i on C!, induces a subgroup tower which t r aps A (k) bu t which is, unfor-

tuna te ly , not of polynomial width. We can reduce the width of the tower to d!, where d

is the degree of X, by consider ing the factor groups A(k)/A (~+1). For these groups we

can show t h a t C k induces a subgroup tower of he ight and width polynomial in IVI.

There are two p rob lems with this smal ler tower: For one , the factor groups A(D/A (k+l)

act on the cosets of A (k+l) and are not p e r m u t a t i o n groups on the ver tex se t of the

graph. We deal with this p rob l em by finding a subgroup A, (k) of Sym(Vk÷l) which is

i somorphic to A(~)/A (k+l). Secondly~ by de t e rmi n i ng a factor group, we seem to lose

an efficient m e m b e r s h i p test . This l a t t e r difficulty can be overcome in part : We find

here t ha t a p e r m u t a t i o n ~ E Sym(Vk÷l) is in A, (k) iff ~ can be ex tended to an au tomor -

ph ism in A (k), which requi res us to solve a ( k + l ) - i somor ph i sm p r ob l e m as will be

explained later . This t echn ique may be applied to ce r t a in subgroups in the tower

t rapping A, (k). If i t were possible to apply it to all subgroups in the tower, t h e n we

would ob ta in a subexponen t i a l i somorph i sm tes t for regular cone graphs of fixed

degree.

We will f irst t rap the group A. (k) in a subgroup tower of a group with known gen-

erators , This subgroup tower will be of polynomial height bu t n o t necessa r i ly of poly-

nomia l width. In order to use the t echn iques of SEction 1 for this tower, we also need

a m e m b e r s h i p t e s t in all groups which arise, and this requi res solving (k+ l ) -

i somorph i sm problems.

We solve the (k+ l ) - i somorph i sm p rob lem by t r ans la t ing i t into a ( k + l ) -

a u t o m o r p h i s m problem. As a consequence , we have a r educ t ion of k - a u t o m o r p h i s m

to ( k + l ) - a u t o m o r p h i s m , and can there fore design a recurs ive p r oc e du r e for

80

deterrninin~ all k-automorphisms. Let ~ e Sym(V~+1) and assume we wish to test

whether ~ e A, (k). -We apply w (more precisely: a simple extension of 71") toX, obtaining

a graph X=. We then consider the disjoint union of X and X~ as the new graph Z. On Z,

we introduce the vertex partition Dk÷ I obtained by first partitioning the components

X and X~ of Z using the partition Ck+1, followed by merging, in the resulting partition,

the pairs of blocks (B, B') where B contains vertices in X and ]3' contains the

corresponding vertices in X=. We then determine the automorphisms of Z which

respect to this partition. Clearly X and X~ are (k+l)-isomorphic if[ one of the genera-

tors just found exchanges them.

To reduce the width of the tower to d!, we insert sufficiently many subgroups into

it. We can show that such subgroups always exist. However, we do not know of a suit-

able membership test for these additional groups and therefore no efficient algorithm

ensues at this time. The insertion of groups without a specific membership test leads

to an interesting open problem which we discuss again in Chapter VI.

We fill in the details into the above outline. First, we establish the isomorphism

between the factor groups A(k)/A (k+1) and permutation groups A. (k) acting on the

vertex sets Vk+ ~, by showing that the eosets of A [k÷1) in A [k) may be characterized by

the action of their members on the set ¥k+i.

THEOm~ 5

Let A(k÷1)a = aA (k+l) be an element of A(k)/A (k+l). If j9 e A(k+l)~, then, for all u e Vk+l,

UP_ - U a,

Proof Note that fl =7a, where 7~A(k+1). Since u 7= u for all UeVk+ I, the

theorem follows. -

Theorem 5 may be considered a proper generalization of Theorem S of Chapter If.

As an immediate consequence of the theorem we have

COROLIAEY Z (Hoffmann)

The elements of A(k)/A (k+1) are in [-1 correspondence with those permutations ~' in

Sym(¥k+1) for which there exists a e A [k) such that, for all u e Vk+1, u = = u ~'.

Note that the permutations ~' form a subgroup A, (k) of Sym(Vk+1) which is iso-

morphic to the factor group. It is this group we wish to determine.

We first show how to trap A, (k) in a subgroup tower of a group with known genera-

tors. Next, we show the existence of additional subgroups which refine the tower to

one of -width d!. Finally, we discuss how to test membership in the subgroups which

81

arise, and, in par t i cu la r , in A, (k).

Let vl, ..., v s be the ve r t i ces in X of d i s t ance k f rom the r o o t v, e n u m e r a t e d lef t to

right. Let El, j be all those ( n o n t r e e ) edges (u,w) of X such t h a t u is a d e s c e n d a n t of v i

and w a d e s c e n d a n t of v i, i.e., u is in the s u b t r e e roo ted in v i and w is in the sub t r ee

roo ted in vj. See Figure 1 1 below.

Ei, j edges

'" ~ d k ve r t i ces n c

Figure I I

We define a sequence of cone graphs Xj = (V,E), I ~ j_< (~)+I, where the edge set

F I consists of all tree edges in X and of all nontree edges (u,w) in E, where u and w are

of distance k or less from the root v. The remaining graphs are defined by

X a = (V, F2) = (V, FIUE~,I)

X 3 = (V,F3) = (V, F~UE12)

Xs+1 = (V, Fs+I) = (V, F~UEI,,)

Xs+ ~ = (V, Fs+z) = (V, Fs+I <JE2 2)

Xs+ 3 = (V, Fs÷3) = (V, Fs+2UEe,s)

Xr = (V, Fr) = (V, Fr-1 UE,,s) = (V,E) = X

We define the groups G (j), I g j ~ r, by letting G (j) = A,(k)(Xj). The groups G (r÷j),

I < j --- s, will be the pointwise stabilizers in G (r) of the sons of v I ..... vj.

The groups G 0), j - (~)+s+l, form the subgroup tower

82

I = G (r+s)< "-" < G (r)< ' ' ° < G 0).

They a re i n d u c e d by the v e r t e x p a r t i t i o n C~. However, s ince we have p a s s e d to f a c t o r

g roups , t he p a r t i t i o n induces t h e s e g roups only in an i n d i r e c t sense . Clear ly t h e

tower has po lynomia l height , We know g e n e r a t o r s for G 0) b e c a u s e of the obvious

L~m% 8

Let V ~ be the se t of all sons of the v e r t e x v~. Then, G O) = lrISym(Y~). i= I

Note that G (r+s) is the trivial group, where r = (~)+I, and that G (r) = A,(k)(X).

Recall that X is of degree d. Consequently, the sets V i are of size at most d. We

would like to apply Lemmata 2 and 3 and conclude that the index of G (j+l) in G (j) is at

most (d!) ~. But this is not possible since we have passed to a homomorphic group,

Le., since the action of A (k) on Vk+ I is not faithful. For a counterexample, consider the

regular cone graph of degree 2 in Figure 12 below:

a b c d e f g h

\ / \ / \ / ' ,/ ! 2 3 4

Figure 1~

Let X (e) be the g r a p h shown, X (5) the g r a p h obtahued by removing the edges in t h e edge

s e t W4, 4. By inspec t ion , A,(~)(X (5)) is g e n e r a t e d by t h e t r a n s p o s i t i o n s (a,b), (c,d), and

(e,f), t hus has o r d e r 8. However, A,(2)(X (8)) = I, h e n c e the group has index 8 > (2!) ~ m

A,(2)(X(5)). I t is no t h a r d to c o n s t r u c t cone g r a p h s of f ixed d e g r e e in which success ive

83

indices are a rb i t r a r i ly large.

We r e d u c e the width of the G-tower to d! by inse r t ing the subgroups H 0'1) . . . . . H 0")

be tween G 0+1) and G(J):

G O+:) = H(J") < H 0's-I) < •. • < H0,O < GO)

Here H 0't) consists of all permutations in GO) which permute the descendants of the

ver t i ces v 1 . . . . . v t e V k such t h a t this p e r m u t a t i o n may be ex t ended to a p e r m u t a t i o n

in G O) .

In g roup- theore t i c t e rms , we ob ta in the subgroups H 0't) be tween G 0) and G 0+1) as

follows: Let K (t) be the pointwise s tabi l izer in G (1) of the de sc e nda n t s of the ver t ices

v 1 . . . . . v t e V k . Since K (t) is n o r m a l in G (0, G0)(~K (0 is n o r m a l in G O) , hence

G0+I)(G0)(~K (t)) is a subgroup of G O) con ta in ing G 0+I). We now see t h a t

H 0'0 = G(J+I)(G0)f~K(0). The index of H (j't+O in H (j't) is a t m o s t d!, s ince the index of

K (t+l) in K (0 is a t m o s t d!. Consequently, we have jus t r e duc e d the width of the sub-

group tower to d! while increas ing its height to 0(sS). Unfor tuna te ly , the re is no

s t ra ight forward efficient m e t h o d for tes t ing m e m b e r s h i p in the groups H (j't), thus we

are unab le to exploit this cons t ruc t i on for designing an a lgor i thm d e t e r m i n i n g the

subgroup tower efficiently.

It r e m a i n s to expla in how to t e s t m e m b e r s h i p in the groups G 0). Call a p e r m u t a -

t ion ~ e Sym(Vk+l) admiss ib le if, for every u e ¥k+1, the ve r tex u and the ver tex u ~

have an equal n u m b e r ot sons of equal d is tance. Admissibi l i ty ensures t ha t the ver-

t ices in the sub t r ee U rooted in u and the sub t r ee W rooted in u ~ are in I-1 cor respon-

dence, and g u a r a n t e e s a n a t u r a l ex tens ion of the p e r m u t a t i o n ~ to the en t i re ve r t ex

set.

DEFINITION 6

Let X = (V,E) be a regular cone graph with root v such tha t the BFS-tree has he ight h.

Let ~ e Sym(Vk+l) be an admiss ib le p e r m u t a t i o n of the ver t ices of d i s tance k + l f rom

v for a fixed value of k < h. Then the s imp le e x t e n s i o n ~ of ?r is the p e r m u t a t i o n of V

defined by

(1) For a l l u e Vj, j ~ k, u# = u.

(~) For all u e Vk+l, u~ = u ~.

(3) Let u < Vj, j > k+ I, be a ve r tex with ances to r w e Vk+l. Let u' be the co r re spond-

ing ve r t ex in the sub t r ee roo ted in w n. Then u # = u'.

84

~ L E 4

Let X be the cone g raph of Figure 8 above, ~r = (2,8) a p e r m u t a t i o n in Sym(V1). Then

the s imple ex tens ion of Tt is ~ = (2,3)(5,7)(6,8).

Given the admiss ib le p e r m u t a t i o n ~ E Sym(Vk+1), we define the graph X~ as the

graph ob ta ined by applying the s imple ex tens ion !P of 7r to X. For example, for

~r = (2,8), the graph X~ is shown in Figure 13 below.

\/ \/ \ /

Figure i3

T.i,:R MA 9

Let # c ~ym(Vk+l). Then ~ ~ A, (k) iff 7r is admiss ib le and the re is a k - i somorph i sm

between X and X~.

Proof Assume t h a t ~ ~ A, (k). Then the re is a ~' c A (k) whose r e s t r i c t i o n to V~+ 1 is

#. Thus ~ is admissible , and the re is a p e r m u t a t i o n X pointwise fixing all ver t ices of

d i s tance k+ 1 or less f rom the roo t such tha t , for the s imple e x t e ns i on 1~ of ~r, # ' = I~X.

Conversely, !e t # be admiss ib le , !~ i ts s imple extension, and X a ( k + l ) -

i somorph i sm f rom X to X~. Then tPX -1 is a k - au tomorph i sm. Since the r e s t r i c t i o n of

~#X -~ t o Vk+1 is ~, i t f o l l o w s t h a t ~r E A , (k). "

We t e s t ( k + l ) - i s o m o r p h i s m by t r an s l a t i ng i t in to a (k+ 1 ) -au tomorph i sm prob lem.

Let X = (V,E) and X' = (V',E') be two cone graphs with roots v and v', respect ively .

We a s sume t h a t V(~V' = ¢. F u r t h e r m o r e , we a s sume t h a t the p e r m u t a t i o n (v,v') is

admiss ib le in the sense of Definition 6 ( imagining the two graphs joined into a bigger

cone g raph with new root r whose two sons are v and v'). Note t h a t X and X' c a n n o t be

(k+ l ) - i somorph ic if (v,v') is no t admissible . Let Z = (VuV', E u E ' ) be the dis joint un ion

85

of X and X'. The partition Dk+ I induces an automorphism group B(k+I)(Z) on Z consist-

ing of all partition respecting automorphisms. Clearly, X and X' are (k+ l)-isomorphic

iff every generating set for B(k+0(Z) contains at least one permutation which

exchanges the X and X' components of Z.

We note that the group B (k+i) can be trapped in a similar subgroup tower as the

groups A (k+0. If X and X' are cone graphs of degree d, then the index of the groups

H (j't) trapping B (k+i) is at most 2.(d!) z. This requires solving, in turn, a (k+2)-

isomorphism problem for membership test in the occurring groups G0). Consequently

we have h e r e a r e c u r s i v e p r o c e s s with h - k levels .

Suppose now t h a t we wish to d e t e r m i n e g e n e r a t o r s for A (k). De te rmin ing g e n e r a -

t o r s for A, (k) involves the m e m b e r s h i p t e s t ou t l ined above, and thus, having t e s t e d

m e m b e r s h i p of e ach g e n e r a t o r ~T of A, (k), we have found a (k+ 1) - i somorphic m a p f rom

X to Xn. Thus, by L e m m a 9, we now have an e l e m e n t ~ in A (k) such t h a t ~bA (~+1) is a

g e n e r a t o r of the f a c t o r g roup A(k)/A (~+l). Then the union (over k) of t h e s e e l e m e n t s

is a gene ra t i ng se t for A (°).

3. p-Groups and Cone Graphs

In th i s sec t ion, we will exp lo re the spec i a l c lass of p-groups and d i scus s r e l a t i on -

ships b e t w e e n these g roups and the a u t o m o r p h i s m g roups of cone g raphs .

For t he c lass of p -g roups , we will develop ef f ic ient c o m p u t a t i o n a l t echn iques .

These t echn iques a re f u n d a m e n t a l to the a lgo r i t hms of s u b s e q u e n t c h a p t e r s and

should be s t ud i ed careful ly .

The m a j o r r e s u l t to be e s t a b l i s h e d is t h a t t he se twise s t ab i l i z e r in a p -g roup can

be found in po lynomia l t ime . The p r e s e n t a t i o n of th is r e s u l t does no t ut i l ize the b e s t

t e chn iques avai lable , and i t will be re f ined in Chap t e r W.

D~UTION 7

A group G is a p-group if eve ry e l e m e n t of G has o r d e r a power of p, where p is a p r i m e

n u m b e r .

In fact , G is a p -g roup iff t he o r d e r of G is pro, m > 0. We wilt see t h a t p - g r o u p s

posses s m a n y s t r u c t u r a l p r o p e r t i e s which a d m i t a r i ch s p e c t r u m of eff ic ient t e c h -

niques . Of spec ia l i n t e r e s t is t he case p=2, which is of p a r t i c u l a r i m p o r t a n c e to t he

r e su l t s of C h a p t e r IV.

86

3. I. Sylow p-Subgroups and Properties oi p-Groups

Lagrange's Theorem (Chapter If, Theorem i) states that the order of a subgroup

of a (finite) group is a divisor of the group order. The converse does not hold: If G is a

group of order n, m a divisor of n, then G need r~o~ have a subgroup of order m, and

there are examples of such cases. The first results to be stated are standard results

from Group Theory, giving conditions under which subgroups of a given order exist

and what their properties are.

THEO~ S (Cauchy)

If the order of a group G is divisible by a prime p, then G contains an element of order

p.

As a consequence, G must have a subgroup of order p, namely a cyclic group of

order p generated by an element of order p.

DEFINITION 8

Let pm m > 0, be the highest power of the prime number p dividing the order of the

group G. Then every subgroup of order pm of G is called a S.VL~, p-s~b~ro~p of G.

The main facts about Sylow p~subgroups are summarized in the following

THEOREM ? (Sylow)

(a) Let pro, m > 0, be the highest power of the prime number p dividing the order of

the group G. Then G contains subgroups of orders pi, i -< i--- m, and each sub-

group of order pi is normal in at least one subgroup of order pi+l, I <- i < m.

(b) All Sylow p-subgroups of G are conjugate in G.

(e) Every subgroup of G whose order is a power of p is contained in at least one Sylow

p-subgroup of G.

(d) if r denotes the number of Sylow p-subgroups of G, then r = I (rood p).

We now summarize results pertaining to elementary properties of p-groups,

DEFTNITION 9

The ce~ of the group G is the subgroup C of G consisting of all elements which com-

mute with every element of G,i.e., C = ~TT c G ] (V~ 6 G)(Tr~ =~T)I.

Note that C may be the trivial group, and that C is always a normal subgroup of G.

In Chapter VI, we will give an algorithm for finding C from generators for G, in polyno-

mial time. The next result asserts that p-groups always have nontrivial centers.

87

THEOREM 8

If G is a p-group, then G has a nontrivial center C, and the index of C in G is divisible

by pC.

An immed ia t e corol lary of Theorem 8 is t ha t every group of order p~ m u s t be its

own center , i.e., is Abelian.

If H is a p-group of degree n, t h e n H m u s t be con ta ined in a t leas t one Sylow p-

subgroup P of S n (Theorem 7c). We will show in Sect ion 3.3 t ha t P can always be con-

s t ruc ted , given gene ra to r s for H. F u r t h e r m o r e , as a consequence of (a), the re has to

be a subgroup tower of P of polynomial height and of width p which t r aps H (Sect ion

3.4). We exploit this fact when comput ing the setwise s tabi l izer in p-groups (Sect ion

~.5).

DEFINITION 10

A subgroup tower

I = G (r)< G (r-~) < • -. <G (°)= G

is a central series for G, if each group G (i) is normal in G, and G(9/G 0+s) is a a sub-

Rroup of the c e n t e r of G / G (i+1). Fu r the rmore , the series is p-step if each factor group

G(i)/G (i+1) is of o rder p.

tf G (i+1) is n o r m a l in G, t h e n it is also n o r m a l in G (9. Note t ha t a group G need no t

possess a cen t r a l series. However p-groups always possess such a series:

T H ~ ; o ~ 9

If G is a p-group of order pro, t h e n there is a subgroup tower

I = G (m) <J G (m-l) <J • " " <I G (0) : G

which forms a p-s tep cen t r a l series.

In par t i cu la r , a Sylow p-subgroup P of S n has a p-s tep c e n t r a l series, and we will

exploit this fact l a te r for t r app ing every subgroup H of P (Sect ion 3,4),

3.2. Wreath Product s and Sylow p-Subgroups of S n

In this sec t ion we explain the s t r u c t u r e of the Sylow p-subgroups of S n. The

s t r u c t u r e is developed convenien t ly in t e r m s of d i rec t and of wrea th products . All

ma te r i a l is s t a n d a r d Group Theory, excep t the r e p r e s e n t a t i o n of these groups as the

a u t o m o r p h i s m group of c e r t a i n cone graphs.

88

Let G < Sym(X) be a p e r m u t a t i o n group of d e g r e e m, H < Sym( t0 a p e r m u t a t i o n

group of d e g r e e n. In tui t ively , the vJreath product GrDH of G b y H is c o n s t r u c t e d as

follows: Take n copies of X, i ndexed b y the po in t s in Y. The e l e m e n t s X ~ G%H are

(n+ I ) - t up l e s ~ y f ~ry 2, ..., ~Yn; ~/), where y~ ~ Y, ~ry i ~ G, and ~p c H. These t u p l e s a c t on

the n copies of X in Lwo s t ages : F i r s t , p e r m u t e Xy~ acco rd ing to 7ryf for e a c h po in t Yi in

Y; t h e n p e r m u t e t he s u b s c r i p t s Yi of the X-copies acco rd ing to ~b.

DEFINITION I 1

Let G < Sym(X) be a p e r m u t a t i o n group of d e g r e e m, H < Sym(Y) a p e r m u t a t i o n group

of d e g r e e n. The wreath produzt, GnoH, of G by H is a p e r m u t a t i o n group of d e g r e e

m.n ac t ing on XxY by

(x,y)X = (XnY,FD,

where x ~ X, y ¢ Y, ~Ty ~ G, ~ ~ H, and X ~ G%H.

Al te rna t ive ly , we visual ize G%H as follows: We draw a t r e e T = (V,E), whe re

V = i h j i l ~ i ~ n , i - < j < - m l O ~ v i i O - < i < - n !

and

E = ! (Vo,Vi) ! l - < i ~ n l U i (v~,lij) I l - < i g n , / - < j - < m l

This t r e e is of h e i g h t 2 and has m~n leaves li, j and n+ t i n t e r i o r v e r t i c e s v i. The r o o t of

T is v o. An e l e m e n t (~I . . . . . ~n; ~) of G~bH p e r m u t e s T by f i rs t apply ing ~i to p e r m u t i n g

the l eaves lid, i . e , by l e t t ing 7r i a c t on ( the leaves of) t he s u b t r e e r o o t e d in vi, followed

by p e r m u t i n g the entiro s u b t r e e s r o o t e d in vi acco rd ing to ~/, The r e su l t ing ac t ion on

the leaves of T def ines G"bH.

Ex ta~I~ 5

Let G = Ss, H = S4. Then C%H is i s o m o r p h i c to the a u t o m o r p h i s m group of the t r e e T

of F igure 14 below.

S~%S4

Figure 14

89

More precisely, the ac t ion of Aut(T) on the se t of leaves of the t r ee is the group GrbH.

The wrea th p r o d u c t is associat ive b u t no t commuta t ive . I t e ra t ing wrea th pro-

ducts cor responds , intui t ively, to building higher t rees . In par t i cu la r , the following is

obvious:

PROPOSITION 1

Let G = S m h O J S m h _ I O j " " " ~JSml , where Smi is the s y m m e t r i c group of degree n-I/, and

let T be a ba lanced t ree of he ight h such tha t every ver tex of d i s tance k - i f rom the

root has exact ly m k sons. Then G consis ts of the ac t ions of Aut(T) on the leaves of T.

Propos i t ion 1 should provide a good geomet r i c in tu i t ion of the n a t u r e of wrea th

products . We now develop the g roup- theore t i c s t r u c t u r e of Sylow p- subgroups of the

symmet r i c group S n.

Let Cp denote the cyclic group of o rder p, p a p r ime n u m b e r . The following is well

known:

THEOR]~ 10 (Kaloujnine)

Let n : alP kl + azp k~ + • - • + arP kr, where p is a p r ime n u m b e r , I -< a i < p, and the k i

are d i s t inc t nonnega t ive exponents . Then every Sylow p-subgroup P of S n is

i somorphic to the d i r ec t p r o d u c t of r groups G i. Each group G i, in t u r n , is i somorphic

to the d i r ec t p roduc t of ai groups H i, and the groups H i are the wrea th p roduc t s of ki

groups Cp. For k i = 0, H i is the tr ivial group.

We use the t h e o r e m to c o n s t r u c t the Sylow p-subgroups of S n as a u t o m o r p h i s m

groups of specific graphs. We begin with the special case p=2.

For S n, we first expand the n u m b e r n in b inary , i.e., n = 2ki+2k~+ • • ' +2 kr, where

the k i a re dis t inct , nonnega t ive exponents . We pa r t i t i on n po in ts in to r blocks

BI . . . . . B r, where B i con ta ins 2 kl points . We then le t the points in B i be the leaves of a

full b ina ry t ree T i of height ki. At this point, we have c o n s t r u c t e d a forest F consis t ing

of r full b ina ry t r ees of d i f ferent heights. We le t G be the a u t o m o r p h i s m group of F,

r e s t r i d t ed to i ts ac t ion on the n leaves in F. Then G is a Sylow 2-subgroup of S n.

~ I ~ 6

Let n=5. We expand 5 in b ina ry as 5 = 22+2 °, and c o n s t r u c t the fores t F of Figure 15

below. Now the group Aut(F), r e s t r i c t ed to the leaves of F, is prec ise ly

IO, (1,2), (3,4), (I,2)(3,4), (i ,3)(2,4), (I,4)(2,3), (1,3,;~,4), (1,4,2,3)I

and is a Sylow 2-subgroup of S 5.

90

i ~ ~ 4 5

A Sylow Z-subgroup of S 5

Figure i5

There are 5 ways of partitioning the points ! I ..... 51 into two blocks of size 4 and i.

For each such partition, there are a different ways of pairing the vertices in the larger

block. Consequently, Ss contains 5-3 = 15 different Sylow 2-subgroups, each iso-

morphic to C~O~C~xl. Note that I5 --- 1 (rood ~). []

In the general case, we can also exhibit a graph whose automorphism group is iso-

morphic to the Sylow p-subgroups of S n, p > 2. Here we use directed graphs, since

the automorphisms of a directed cycle of length p naturally correspond to the action

of Cp. Note that Cp is isomorphic to the automorphism group of the cone graph Tp in

Figure 16 below, restricted to the action on the leaves.

. ~ p vertices

% Figure 16

tn the case p=~, Te can be a tree since Ce = $2. So for the k-fold wreath product

of C 2 a full binary tree may be used. For the general case, we have to build a regular

(directed) cone graph of height k from the graphs T 9. It is not hard to see that the

automorphism group of this cone graph, acting on the set of leaves, is the k-fold

wreath product of Cp. Figure 17 below shows the graph for a Sylow 3-subgroup of $I~.

Observe that for p > ~ this construction may result in a graph with more than one

cone graph of height k, since in Theorem 10 ai may be larger than 1. Here it is impor-

tant to realize tha t we must consider only those automorphisms which fix the root of

each component cone graph, since we construct a direct product of i terated wreath

products. For p = 2 this remark is vacuous since all t rees in the binary forest neces-

sarily have distinct heights.

91

I-~2-'3 4~'*~-~6 7-"8"-*9 tO--,11-,.12

A Sylow S-subgroup of $15

Figure 17

13--,14 -~15 \V

3.3. I m p r i m i t i v i t y of p-Groups

By Theorem 7c, if G is a p-group of degree n, t h e n G is con t a ined in a t leas t one

Sylow p-subgroup P of Sn. We now cons ider the p r ob l e m of finding the group P given

G:

PROBLEH 3

Given a gene ra t ing se t for a p-group G of degree n, p a fixed p r ime n u m b e r , find a

gene ra t ing se t for a Sylow p-subgroup P of Sn which con ta ins G as a subgroup.

An efficient a lgor i thm for P rob lem 3 will be useful as a first s tep towards an

efficient a lgor i thm for finding setwise s tabi l izers in p-groups. This in t u r n will play a

role in devising an i somorph i sm test for graphs of fixed valence, and also for cone

graphs of fixed degree.

An i m p o r t a n t p rope r ty exploited when finding P is the impr imi t iv i ty of p-groups:

Let G < Sym(X) be a p e r m u t a t i o n group, and suppose X can be pa r t i t i oned into dis-

jo in t blocks X I . . . . . X r, such tha t every e l e m e n t of G e i ther s tabi l izes X i setwise, or

maps all points of X i to points of Xj, 1 -< i, j -<- r. tf this pa r t i t i on is nontr ivial , i.e., if

r ~ I and r ~ IXl, t hen the pa r t i t ion is called a systerr~ of irnprimitivity for G. The

blocks Xi are called sets of irnprirnitivity. A permutation group G is imprimitive if

there is a sy s t em of impr imi t iv i ty for G. Otherwise G is primitive.

For example, Sn is a pr imi t ive group, whereas S~%Sk is impr imi t ive with a s y s t e m

of impr imi t iv i ty consis t ing of k blocks of size i each. In par t i cu la r , if G is an in t r ans i -

tive group (cf. Chapter II, Definition 8 ft.), t hen the orbi t pa r t i t ion of the p e r m u t a t i o n

domain cons t i t u t e s a sy s t em of impr imi t iv i ty for G, thus every in t rans i t ive group is

92

impr imi t i ve . Note t h a t an i m p r i m i t i v e g r o u p m a y have d i f fe ren t s y s t e m s of i m p r i m i -

t ivity.

The following t h e o r e m s u m m a r i z e s some of the s t r u c t u r a l p r o p e r t i e s of i m p r i m i -

t ive b u t t r a n s i t i v e g roups .

T H E O ~ 11

Let G < S n be a t r ans i t i ve bu t impr imi t i ve group, and l e t Y be a s e t of i m p r i m i t i v i t y

f o r G , x c Y . Then

(a) The s t a b i l i z e r G x of x in G is a p r o p e r s u b g r o u p of t h e se twise s t ab i l i z e r Gy of ¥ in

G, and C~ is a p r o p e r s u b g r o u p of G.

(b) Each se t of i m p r i m i t i v i t y (in the s y s t e m conta in ing Y as b lock) con ta ins e x a c t l y

(G¥:Gx) points , and t h e r e a r e (G:Gy) d i f f e ren t s e t s of i m p r i m i t i v i t y in t he sy s t em.

Conversely , l e t G < Sn be a t r an s i t i ve p e r m u t a t i o n group, and l e t Gx be the s t ab i l i ze r

of the p o i n t x in G. If t h e r e is a p r o p e r subg roup H of G which p r o p e r l y con ta ins G x,

t h e n

(c) G is i m p r i m i t i v e and one of i t s s e t s of i m p r i m i t i v i t y is t h e o r b i t Y of x in H.

(d) G has a s y s t e m of i m p r i m i t i v i t y cons i s t ing of (G:H) blocks , among t h e m Y, and

e a c h b l o c k c o r r e s p o n d s to a r i gh t c o s e t of H in G.

TnEOR~X 12

Let G be a t r an s i t i ve p -g roup of deg ree p k k > 1. Then G p o s s e s s e s a s y s t e m of

i m p r i m i t i v i t y cons is t ing of pk-1 se t s of impr imi t iv i ty , e a c h of size p.

If G is a Sylow p - s u b g r o u p of t he s y m m e t r i c g roup , t h e n t h e se t s of i m p r i m i t i v i t y

of T h e o r e m 12 a re t he s e t s of b r o t h e r s among the leaves of t he a s s o c i a t e d cone g raph .

We outline the ideas in constructing a Sylow p-subgroup P of S n containing the

given p-group G of degree n.

Recall Theorems 7 and 10. If G is intransitive, then, using Algorithm4 of

Chapter II, we split the permutation domain into the orbits Bi of G (I ~ i -< s). Note

that each orbit B i must be of length pk, k -> 0, for otherwise the order of G cannot be a

power of p (see Chapter If, Theorem 3). So, let Wi be the transitive constituent of G

obtained by restricting the action of G to the orbit B i, and note that W i is again a p-

group. Recall that G is a subgroup of the direct product G' of its transitive

constituents, again a p-group.

We proceed in two stages: First, for each constituent p-group Wi, we determine a

93

Sylow p - s u b g r o u p Pi of Sym(Bi) con ta in ing Wi as subgroup . P r o c e e d m g recu r s ive ly , we

e s sen t i a l l y bu i ld the a s s o c i a t e d (d i r ec t ed ) r e g u l a r cone g r a p h of deg ree p whose au to -

m o r p h i s m group, when r e s t r i c t e d to the leaves of the g raph , is Pi. Second , if in t he

r e su l t ing co l lec t ion of cone g r a p h s t h e r e a r e m o r e t h a n p - 1 g r aphs of he igh t k, t h e n

p of t h e m a re chosen a r b i t r a r i l y and c o m b i n e d into a cone g r a p h of he igh t k+ 1. This

s t ep wil l have to be r e p e a t e d unti l , for each i n t ege r k, t h e r e a r e less t han p cone

g r aphs of he igh t k. The final co l l ec t ion of cone g r a p h s now d e t e r m i n e s a Sylow p-

subg roup of S= which m u s t con t a in G as subgroup .

Recal l t h a t the cone g r aphs a re bu i l t up f rom the g r aphs Tp of Sec t ion 3.3. This

m e a n s tha t , a t each level, we m u s t join the se t s of b r o t h e r s in to a d i r e c t e d cyc le of

l eng th p. During s t age one of the c o n s t r u c t i o n i t is c ruc i a l to l ink up b r o t h e r s in t he

c o r r e c t o rde r . Clear ly th is o r d e r can be d e t e r m i n e d quickly f rom the g e n e r a t i n g set .

During s t age two, the cycl ic o r d e r of b r o t h e r s m a y be c h o s e n a rb i t r a r i l y , s ince G ac t s

i n t r ans i t i ve ly on the leaves of t he cone g r a p h s to be combined . We i l l u s t r a t e t h e two

s t ages with

EXAmPI~ 7

Let p=2, n = t 0 , and a s sume t h a t G = <(1,2), (3,4), (1,5)(2,6), (7,8)>, a 2-group. C is

i n t r ans i t i ve and has the orb i t s ~I,2,5,6~, t3,41, ~7,8~, 191, /10~. I ts t r ans i t i ve cons t i -

t u e n t s a r e the 2-groups W i = <(1,2), (1,2)(5,6)>, ~ = <(3,4)>, W 3 = <(7,8)>,

W 4 = <(9)> = I, W~ = <( I0 )> = I. For W 1 we ob ta in a b i n a r y t r e e of he igh t 2, r e p r e s e n t -

ing P1, a Sylow 2-subgroup of Sym(t l ,2 ,5 ,61) . For W~ and W3 we o b t a i n b i n a r y t r e e s of

he igh t 1, and for t h e r e m a i n i n g c o n s t i t u e n t g roups we ob ta in t r e e s of he igh t 0. We

now have one t r e e of he igh t 2, two t r e e s of he igh t 1, and two t r e e s of he igh t 0. F r o m

the two t r e e s of he igh t 0 we bui ld a new t r ee of he igh t 1, so we now have t h r e e t r e e s of

he igh t 1. F r o m two of t h e m we bui ld a new t r e e of he igh t 2, which is t h e n c o m b i n e d

wi th the o t h e r t r e e of he igh t 2 in to a new t r e e of he igh t 3. The r e su l t i ng fo r e s t is n o t

unique, b u t the occu r r ing t r e e he igh t s are. One poss ib le final f o r e s t is shown in Fig-

ure 18 below. []

Note t h a t g e n e r a t o r s for P can be found by i n spec t i ng the c o n s t r u c t e d r e g u l a r

cone g r a p h fores t .

The nont r iv ia l s t ep in th is c o n s t r u c t i o n is the d e t e r m i n a t i o n of a Sylow p-

subg roup P conta in ing a t r an s i t i ve p-group. We now d e s c r i b e th is p a r t in m o r e de ta i l .

94

! 2 5 S 7 8 3 4 9 10

A Syiow 2 - subgroup of Sao containing G

Figure 18

Recal l t h a t t he d e g r e e n of the t r ans i t i ve p -group G m u s t be a power of p. We d is t in -

guish t h r e e cases : n = l , n=p , n>p. For n = t , we c o n s t r u c t a cone g r a p h of he igh t 0,

cons is t ing of only one ver tex . For n=p, we know t h a t G is the cycl ic g roup of o r d e r p,

and the cone g r a p h Tp of Sec t ion 3.3 suffices. Note he r e t h a t G is a l r e a d y a Sylow p-

subgroup of the s y m m e t r i c group of degree p.

For n = pk k > i, there must exist a system of h~primitivity for G consisting of

p r e c i s e l y pk-J b locks of size p (Theorem 12). These b locks will c o r r e s p o n d to t h e

leaves in t h e s u b t r e e s of h e i g h t i of t h e cone g r a p h we wish to c o n s t r u c t . Here we

p r o c e e d r ecu r s ive ly : We f i r s t find such a s y s t e m of i m p r i m i t i v i t y for G. We t h e n con-

s t r u c t a g roup G' h o m o m o r p h i c to G by cons ide r ing t h e a c t i o n of G on the s e t s of

impr imi t iv i ty . Note t h a t G' i s a t r an s i t i ve p -group of d e g r e e pk-1. We find g e n e r a t o r s

for the group G' from the generators for G and the required system of imprim/tivity.

Briefly, we enumerate the n k-I sets of imprimitivity, For each generator ~r e Spk we

construct a generator ~'< $9k-i by inspecting how the sets of imprimitivity are

mapped. The resulting set K' generates G'. Proceeding with G' inductively, we deter-

mine the deeper levels of the cone graph. The reeursion ends after exactly k stages.

Note that we have just reduced Problem 3 to the following

PROBLEM 4

Given g e n e r a t o r s of a t r an s i t i ve p -group G of deg ree n = p k k > 1, find n_ se t s of P

i m p r i m i t i v i t y of size p for G, and find g e n e r a t o r s for G', t he g roup of d e g r e e pk-1 of

the actions of G on the sets of imprimitivity.

95

We now turn to finding the required set of imprimitivity in polynomial time. The

a lgo r i t hm to be d e s c r i b e d m a y be used to d e t e r m i n e , in po lynomia l t ime , w h e t h e r an

arbitrary transitive permutation group G is imprimitive. Note that Algorithm 4 of

Chapter II may be used to test whether G is transitive, also in polynomial time.

The centerpiece of the algorithm is a procedure for determining the smallest set

of imprimitivity for G containing the points 1 and i in the permutation domain. Now if

G is a transitive primitive permutation group of degree n, then, for 2 ~ i-n, the

smallest set of imprimitivity containing both i and i must be the entire permutation

domain. On the other hand, if G is transitive and imprimitive, then there exists at

least one value of i for which the smallest set of imprimitivity containing both I and i

is a proper subset of the permutation domain. Therefore, with at most n-1 invoca-

tions of the procedure we can test whether G is imprimitive, and if so, find a nontrivial

system of imprimitivity for G.

We describe the procedure for finding the smallest set of imprimitivity eontainill~

both i and i. This procedure is a classical application of the disjoi~%t set u~or~/~%d

atgor/t/%m (see Section 4).

Let E = IB I ..... Bsl be any partition of the permutation domain Ii ..... n~. Begin-

ning with the trivial partition E 0 consisting of n singletons, the object is to determine

a partition E I which is a system of imprimitivity for G in which both i and i are in the

same block.

Throughou t t he c o m p u t a t i o n , we will m a i n t a i n a c u r r e n t p a r t i t i o n E' of the p e r -

m u t a t i o n d o m a i n and a s t a c k of pa i r s (u,v). The func t ion of the s t a c k e d pa i r s is to

ensu re t h a t the po in t s x and y, con t a ined in some block B of E', a r e m a p p e d into the

s ame b lock B' of E' by every g e n e r a t o r of G (e.g., u = x =, v = y~ for some g e n e r a t o r n).

This m a y r e q u i r e merg ing d i s jo in t b locks in E' and s t a c k i n g new pa i r s . Eventual ly , t he

s t a c k is e m p t i e d , and the final p a r t i t i o n will be the d e s i r e d s y s t e m of impr~mit ivi ty .

We p e r f o r m two ope ra t i ons with b locks in the p a r t i t i o n E and with po in t s in t he

p e r m u t a t i o n domain : find(x) d e t e r m i n e s which b lock in E con ta ins the po in t x;

un ion(x ,y ) m e r g e s t he (dis joint) b locks t3 and B' in E con ta in ing the po in t s x and y,

r e spec t ive ly . The s p e e d of the ' a l go r i t hm d e p e n d s c ruc ia l ly on the i m p l e m e n t a t i o n of

t h e s e ope ra t ions . Briefly, we will r e p r e s e n t the b locks as i n v e r t e d t r ee s , with the

po in t a t the roo t serving to iden t i fy the block. Two b locks a re m e r g e d by adop t ing

the t r e e r e p r e s e n t i n g the sma l l e r b lock as s u b t r e e of t h e r o o t of t he o t h e r t r ee , t ies

96

b r o k e n a r b i t r a r i l y . The o p e r a t i o n find(x) has to t r a v e r s e t he p a t h f rom the p o i n t x to

t he roo t of t he t r e e con ta in ing x. Here we use p~.th compression, i.e., x and eve ry

po in t y e n c o u n t e r e d in the t r a v e r s a I a r e s u b s e q u e n t l y m a d e sons of the root . With

this m e t h o d of i m p l e m e n t a t i o n , i t is well known t h a t a s equence of 0(n .m) u n i o n s and

f inds m a y be e x e c u t e d in O(n-m-log2 (n)) s t eps , w h e r e the value of log2*(n) is the

s m a l l e s t i n t e g e r k such t h a t [log2k(n)] = 1. | !

ALGORITI~ 2 (Se t of I m p r i m i t i v i t y )

Input

Output

Comment

Method

i. begin

Genera t i ng se t KcS n of the t r an s i t i ve g roup G, and po in t i, 2 -< i -< n.

Equivalence p a r t i t i o n E of l i . . . . . nl i nduced by the s m a l l e s t s e t of

i m p r i m i t i v i t y for C con ta in ing bo th i and i.

Note t h a t E m a y con ta in only one c lass of size n.

2. In i t ia l ize E to con t a in n s ing le ton se ts ;

3. In i t ia l ize STACK to con t a in the pa i r (1,i) only;

4. wh i l e STACK ~ e m p t y do beg in

5. u n s t a c k the pair (x,y);

6. if f ind(x) ~ find(y) t h e n b e g i n

7. union(x,y) in E;

8. f o r e a c h ~ c K do

9. s t a c k (x~,y~);

I0. end;

i i . end;

13. output(E);

i3. end.

L~MA 10

Algor i thm 2 t e r m i n a t e s .

P r o o f Observe t h a t a p a i r (u,v) is s t a c k e d iff two n o n e m p t y d i s jo in t b locks in E

are m e r g e d . Thus, the whi le- loop (Lines 4-11) is e x e c u t e d a t m o s t ( n - l ) . IKI t i m e s . -

97

We nex t prove t h a t Algor i thm 2 d e t e r m i n e s a s y s t e m of impr imi t iv i ty for G in

which both 1 and i are in the same block. The following l e m m a asse r t s t ha t i t suffices

to cons ider only the mapp ings of the blocks provided by the g e n e r a t o r s of the group,

and is obvious.

LEsSA 11

Let G = <K> be a group of degree n, E = ~B1 . . . . . Bs~ a pa r t i t i on of the p e r m u t a t i o n

domain. Then this pa r t i t i on is a sy s t em of impr imi t iv i ty for G iff, for each ~ e K,

e i ther s tabi l izes the poin ts of Bi setwise, or maps every point of Bi into poin ts of the

same block Bj, 1 -< i ~ s.

For a~y pa r t i t i on E, le t us call two points x and y E-equivalent, x --E Y, if x and y

are in the same block of E. We will es tabl ish

TI--I]gOEF_,M 13

When reaching Line 11 of Algori thm 2, the following asse r t ion (L) is t rue:

(L): If z -~E w and the re is a g e n e r a t o r Tr e K such t h a t z ~ ~-~ w n, t h e n Lhere are pairs

(Ul,Vl) . . . . . (Ur,Vr) in STACK, such tha t z n -E ul, vl -=E uz .. . . . vr ~-E W ~.

Proof Let us call the cha in of pairs in asse r t ion (L) an equivalence chain.

Observe t ha t (L) is t rue when first en t e r ing the while-loop. Thus, i t suffices to show

tha t (L) r ema ins t rue af ter execut ing Lines 5-10. So, le t E be the par t i t ion , STACK the

gtack immediaLely before execuLing Lines 5-10, and let E' and STACK' be the pa r t i t i on

and s tack resu l t ing from execut ing these lines. We assume (inductively) t ha t E and

STACK satisfy (L), and t h a t Line 5 removes the pair (x,y) f rom STACK.

Let z -E' w such t h a t z n ~ , w n, for some ~ E K.

Case (1): z---E w. By a s s u m p t i o n STACK conta ins an equivalence cha in (ul,vl) ,

.... (ur,Vr) for z ~ and w ~. If (x,y) does no t occur in this chain, t hen STACK' will still con-

t a in the chain. Otherwise, le t (x,y) be the pair (ui, vi). Then (ul,vl) . . . . . (ui_l,vi_l) ,

(ui.l,vi+l) . . . . . (Ur,Vr) is in STACK' and is an equivalence chain for z ~ and w ~, since now

X -=E' Y.

Case (2): z ~-Ew, Then the block B of E' containing both z and w must be the

union of the block B x containing both x and z, and the block By containing both y and

w in E.- By Case (1) above, t he re exists an equivalence chain (ul,vi) . . . . . (ur,Vr) for z"

and x =, and an equivalence chain (ur+l,vr+l) . . . . . (ut,vt) for y" and w ~ in STACK', no t con-

ta in ing the pai r (x,y). Clearly t hen (u~,vl) . . . . . (ur,vr), (x~,yn), (ur+t,Vr+l) . . . . . (ut,vt) is an

equivalence chain for z ~ a n d w ~ and is in STACK'. -

98

Now we obta in the i m m e d i a t e

C o R o ~ 3

Algori thm 2 d e t e r m i n e s a sy s t em of impr imi t iv i ty for G.

Proof By Lemma i0, the a lgor i thm t e r m i n a t e s with an empty s l ack and a par t i -

t ion El, which, by L emma t 1 and Theorem 13 is a s y s t e m of impr imi t iv i ty for G. "

THEORF~ 14 (Atkiuson)

Algori thm 2 is correc t .

Proof We need to prove t h a t in the pa r t i t i o n Ef d e t e r m i n e d by the a lgor i thm the

block B conta in ing t and i m u s t be the smal le s t se t of impr imi t iv i ty for G conta in ing

both points. This is done with a straightforward induction proving that every partition

E at the time of reaching Line ii is a refinement of the partition in which the block

containing both i and i is minimal. •

Let us analyze the running time of [he algorithm. Since no more than n-i unions

of disjoint sets are possible, Lines 7-9, nested deepest in the algorithm, cannot be

activated more than n-i times. Thus, the total time spent in Lines 7-9 is 0(IKl.n).

(Recall that a union instruction requires constant time.) The remaining work of the

algorithm is proportional to the number of pairs stacked, neglecting the cost of the

iind instructions. Clearly, no more than (n-i).]K[ pairs are stacked. We execute

O(IKl'n) find instructions. Here we know that the total time required is

0(IKI "n'log2*(n)), which dominates the running time. In summary, we have

THEOREM 15 (Hoffmann)

Let C = <K> be a p e r m u t a t i o n group of degree n. Then in a t m o s t 0(IKI-n2.1og2*(n))

s teps we can d e t e r m i n e whe ther G is impr imi t ive , and if so, find a nont r iv ia l sys tem of

imprimitivity for G.

Proof Using Algorithm 4 of Chapter If, we determine first in O(IKI'n) steps

whether G is transitive. If not, then G is imprimitive and the orbit partition is a non-

trivial system of imprimitivity. Next~ if G is transitive, by at most n-i invocations of

Algorithm 2 we can determine whether G is imprimitive and find a nontrivial system

of imprimitivity. Thus the stated worst case time bound is correct. -

COROLLARY 4

If G = <K> is a transitive p-group of degree n, n > p, then in O(IKl.n~-log~*(n)) steps

we can determine a system of imprimitivity for G consisting of exactly n_n_ blocks of P

size p and find a generating set K' of size at most !K 1 for G', where G' is the action of

G on the se t s of i m p r i m i t i v i t y found.

Proof Obvious. -

99

We have jus t solved P r o b l e m 4 in po lynomia l t ime , and we now cons ide r the t ime

r e q u i r e d to solve P r o b l e m 3.

If G is a t r an s i t i ve p-group , t h e n c l ea r ly t he con ta in ing Sylow p - s u b g r o u p P and i ts

a s s o c i a t e d cone g r a p h can be d e t e r m i n e d by r e p e a t e d a p p l i c a t i o n of Coro l la ry 4. If G

has d e g r e e n = ph t hen the c o n s t r u c t e d cone g r a p h has he igh t h. At level i in t he

g raph , we c o n s i d e r a t r ans i t i ve p -group of deg ree n Thus, we t ake no m o r e t h a n pl

n 2 • n c' I KI ' ~-~-log s ( ~ s t e p s to d e t e r m i n e the r e q u i r e d se ts of impr imi t iv i ty , where c is a

c o n s t a n t i n d e p e n d e n t of n, p, and i. In the s a m e t ime bound we can c o n s t r u c t f rom K

a new se t of g e n e r a t o r s for the g roup ac t ion on the se t s of impr imi t i v i t y . Thus, we

find the cone g r a p h in no m o r e t h a n

h n 2 • n s

0 ( E (IKl' i=0 ~ l ° g s ( ~ ~ ) ) - < 0(IKI 'nS"l°g2*(n)" p 2--sp-~i-i ) - < 0(IKI 'nS"l°gs*(n)'2)

s teps , s ince p -> 2. Observe t h a t we can c o n s t r u c t the g e n e r a t o r s for P in the s a m e

t ime bound.

In the case where G is impr imi t ive , we f i rs t sp l i t G in to i ts t r an s i t i ve cons t i t uen t s .

This is done in 0 ( IKI -n ) s t e p s using Algor i thm 4 of C h a p t e r II. Then, for e ach cons t i -

tuen t , we d e t e r m i n e the c o r r e s p o n d i n g g roup P and the a s s o c i a t e d cone g raph . Since

these g roups ac t on d is jo in t p e r m u t a t i o n domains , the en t i r e c o n s t r u c t i o n can also be

done in 0( I KI 'nS'log2*(n)) s teps . Having c o m p l e t e d this pa r t , i t m a y be n e c e s s a r y to

combine r e p e a t e d l y p cone g raphs of he igh t h in to a new cone g r a p h of he igh t h + i .

Clear ly th is can be done in t he s t a t e d t ime bound. In s u m m a r y , we t h e r e f o r e have

COROLIAI~ 5 (Hoffmann)

If G = <K> is a p-group, t hen a g e n e r a t i n g se t for t h e Sylow p - s u b g r o u p con ta in ing G

can be found in 0(IKI-nS.loga*(n)) s teps .

This r e s u l t solves P r o b l e m 3 in po lynomia l t ime.

3.4. The Central Series

100

In Sec t ion 3.3, we have shown how to e f f ic ien t ly find a Sylow p - s u b g r o u p P of Sn

conta in ing as s u b g r o u p a g iven p -g roup G of d e g r e e n. We will now show how to con-

s t r u c t a subg roup tower which t r a p s the group G and m a k e s i t ( k , c ) - acees s ib l e f rom

P. The o b j e c t of th is c o n s t r u c t i o n is to r e d u c e the p r o b l e m of finding the se twise s t a -

b i l izer in a p -g roup G to the case where G is a Sylow p - subgroup of the s y m m e t r i c

g roup . The a c t u a l r e d u c t i o n will follow f rom Theorems 11 and 14 of C h a p t e r II.

The t e c h n i c a l tool u sed in t r a p p i n g the s u b g r o u p G of P will be the c o n s t r u c t i o n of

a p - s t e p c e n t r a l se r i e s for P (el. Defini t ion 10). We beg in with the de r i va t i on of th is

ser ies , and c o n s i d e r

PROBLEM 5

Given a Sylow p-subgroup P of S n of order pr r > 0, determine a sequence of r ele-

ments of P, 91 ..... ~r, such that the groups G (r-i) = <3# I ..... ~i>, 0 <- i-< r, form a p-

step central series for P.

Theorem 9 asserts that such a sequence always exists. We will construct this

sequence recursively, imitating the decomposition of P in terms of cyclic groups Cp,

direct products, and wreath products. Note that this decomposition is available as

part of the construction of P in Section 3.3. In particular, we will find a generating

sequence consisting only of permutations of degree p.

We begin with the cases P = Cp and P = PIxP2. The following is obvious:

[ ,m~A 12 ( Let P = Cp, ~ = ~l,2,...,p;. Then I#i = ~ determines a p-step central series for P.

Observe that ~ is of order p. The next result is equally straightforward, and is a

consequence of the properties of direct products.

L ~ 13

Let P = PI×P2. Assume that ~i ..... ~, determines a p-step central series for PI and

t h a t ~/s+t . . . . . •r d e t e r m i n e s a p - s t e p c e n t r a l se r i e s for P~. Then ~Pl . . . . . ~s, ~s+1 . . . . . ~r

determines a p-step central series for P.

Of course, the sequence ~s+1 ..... ~r, ~i, -.', ~s determines another p-step central

series for P.

The nontrivial step in the Construction to be given is how to handle the wreath

101

p roduc t P1rbCp, where PI is a p-group for which we a l ready have found a p-s tep cen-

t ra l series. Here we find the following observa t ions helpful:

Let B be a group of degree n, A a group of degree m, and cons ider G = A%B.

Recall t ha t e l emen t s in G are (n+ l ) - tuples whose first n c o m p o n e n t s are e l e m e n t s in

A, and whose n + l st c o m p o n e n t is an e l e m e n t of H. Let X = (al , a2 . . . . . an; g) and

= (71, 72 . . . . . Yn; 6) be two e l emen t s in G. Then the i r p r oduc t is the (n+ 1)-tuple

x~ = (alylp, aeTep . . . . . anYn~; f16)

F u r t h e r m o r e , the following is clear:

LgsxA 14

Let C = A%H, where B is a p e r m u t a t i o n group of degree n. Then G conta ins a no rma l

subgroup H isomorphic to the n-fold d i rec t p roduc t of A with itself. The e l emen t s of H

are precise ly those ( n + l ) - t u p l e s in G whose last c o m p o n e n t is the ident i ty , F u r t h e r -

more, the factor group G / H is i somorphic to H.

Observe t ha t a p e r m u t a t i o n ~ c o m m u t e s with every e l e m e n t of a group G = <K>

iff ~ c o m m u t e s with every gene ra to r in K. Moreover, if P = Pl%Cp, and P1 = <K>,

t h e n KUIX~ is a gene ra t ing se t for P, where X = (0 . . . . . 0; ~T), and 7r = (1,2 ..... p). Note

t ha t X has order p.

Given the p-group P1 with a p-s tep cen t r a l series d e t e r m i n e d by ~1 . . . . . @r, p e r m u -

ta t ions of o rder p, we will find gene ra to r s for a p-s tep c e n t r a l ser ies for P = PI%Cp.

The length of the sequence we seek m u s t be p.r+ 1, since P has order IP11P'P,

Let H be the no rma l subgroup of P isomorphic to the p-fold d i rec t p r oduc t of PI

with i tself (cf. Lemma 14). We will c o n s t r u c t a sequence ~1,1 . . . . . ~l,p, ~2,1 . . . . . ~r,p, X

which de t e rmines a p-step cen t r a l ser ies for P. Note t ha t this sequence has the

cor rec t length. The p e r m u t a t i o n X will be as above, and we will d e t e r m i n e the p e r m u -

ta t ions ~i,j as e l emen t s of H. To obta in the sequence, we use p mapp ings hl . . . . . h~ of

P1 into H. We define

h~(~) = ( ~ . 1 ~gi.2 . . . . . ~i ,p; 0) ,

and we d e t e r m i n e the exponen t s gi,j next.

There are two aspects to the der iva t ion of the gtj: One, we have to s tudy how a

p e r m u t a t i o n hi(3b ) can c o m m u t e with the new g e n e r a t o r X, so as to ob ta in a cen t r a l

series; second, we want successive factor groups in the series to have order p, so t ha t

t02

we obtain a p-step central series.

In light of our second concern, we insist that gi, i = 1 and gid = 0, j < i. It is then

obvious that a permutation ~ of order p is mapped to hi(~), which is also a permuta-

tion of order p. With respect to the first point, we insist that

(S1) hl(tk)X = xh1(~)

and, for i s i < p and for I# of order p,

= , h i ~ ) j xhi+1(~)

~e will determine the remaining exponents gi,j under these assumptions. For I~ E PI,

we have

and

xh1(1) ) = (~/g1.~ ..... ~,g1,~ I#; .~)

From (Sl) we obtain glj = I, I -< j ~ p. Now the following is clear:

I ~ 15

G (r-l'l) = <hl(~>l)> has order p and is in the center of P = PlcbCp.

Proof We already have shown that hl(!#l) = !#i, I commutes vdth X- It must also

commute with every element of H, since 91 commutes with every element of PI and H

is a direct product of these groups. -

Recall our assumption that gi,i = I, gij = 0, j < i, and consider the products

hi+l(9)X = (0 . . . . . O, q/, .~gi+l,i+z . . . . . .¢gi+l,p-1, .¢si+i,p; ~)

and

(hi(9))- lxhi+!(~) = (0 . . . . . O, ,~,gi+Li+z- ai,i+l ~ i+ l , i+3- ai,i+~ . . . . . ,lO~i+l,p -ai ,p-1 ~-gi,p; ~.)

Observing ($2), we obtain the recurrence

(RI) gi+i,k = gi+1,k-i + gi,k-1 i <--- i < p, i+1 < k ~ p

and the equation

(R2) gi+1,p + gi,p = 0

103

We t h e r e f o r e def ine

k - 1 -- ( k - i ) ,

where we a s s u m e (k) = 0 w h e n e v e r j < O. Clear ly th i s d e f i n i t i o n sa t i s f ies (Rt ) a n d is

c o n s i s t e n t wi th t he p r e v i o u s d e f i n i t i o n of h 1 a n d wi th t h e e a r l i e r a s s u m p t i o n s a b o u t

gi,j. We now have

LM~MA 16 ( H o f f m a n n )

If ~ ~ PI has o r d e r p, t h e n hi+l(~) X = (hi(q]))-Ixhi+l(~).

P roo f S ince the gi,j sa t i s fy (R1), we only have to show t h a t (R2) ho lds . Obse rv ing

t h a t ~ has o r d e r p, we t h u s have to show t h a t (ppT l t )+ (Pp- -~ ) = (pP_i) is c o n g r u e n t to

0 m o d u l o p. S ince 1 -< i < p, t h e d e n o m i n a t o r of (pP-i) c o n t a i n s on ly f a c t o r s s m a l l e r

t h a n p. S ince p is p r i m e , ( P _ i ) is d iv is ib le b y p a n d t h e r e f o r e c o n g r u e n t to 0 m o d p. i

We def ine t he s e q u e n c e ~Pi,j, 1 <- i-< r , i -< j -< p, by ~Pi,) = hj(~i)- F u r t h e r m o r e , we

def ine G(r-i'9 = <~P1,1 . . . . . ~l,p, ~g,1 . . . . . ~i-l ,p, ~i,I . . . . . ~i4 >, and

G (r-9 = <¢1 . . . . . ~i>. We will p rove t h a t

I ~ G (r-l 'O ~ - • - ~ G (r-1'1~) ~ G (r-~l) ,~ - • • ,~ G (0'I~) ~ P

is a p-step central series for P = PI%Cp, provided that

I<~G (r-1)¢~ - . , <~G ( ° ) = P 1

is a p - s t e p c e n t r a l s e r i e s for P b a n d the q/i a r e p e r m u t a t i o n s of o r d e r p.

T.~:MMA 17

Let 3~1 . . . . . ~ r be p e r m u t a t i o n s of o r d e r p d e t e r m i n i n g a p - s t e p c e n t r a l s e r i e s for P1. If

G (r-i'j) is a n o r m a l s u b g r o u p of P, a n d if G (r-i+1'p) is t h e d i r e c t p r o d u c t of p cop ies of

G(r-i+l) = <~Pl . . . . . ~i_1>, t h e n GCr-id+l)/G (r-i'j) is in the c e n t e r of P / G (r-IJ) a n d is of

o r d e r p. F u r t h e r m o r e , G (r-i'j+l) is also a n o r m a l s u b g r o u p of P,

Proof By L e m m a 16, ~/i,j+~ c o m m u t e s wi th X m o d u l o G (r-~j). Let

= (~1 . . . . . ~p; 0 ) E H. Then ~k E P1. S ince ~i c o m m u t e s wi th ~k m o d u l o G (r-l+l), a n d

s ince G (r-i+l,v) is t h e d i r e c t p r o d u c t of p copies of G ff-i+O, the e l e m e n t s ~'i,j+l a n d

c o m m u t e m o d u l o G (r-i+l'p). BUt G (r-i+l'p) is a s u b g r o u p of G (r-i'j), t h u s G(r-i 'J+l)/G (r-i ' j)

is a s u b g r o u p of the c e n t e r of P / G (r-i'j).

~04

Next, tot re ~ G (r-i'j~l). Observe t h a t the @i,I . . . . . ~/~9 c o m m u t e with each other .

Since G (r-~+i) <~ C (r-~), and s ince G (r-i+1'p) is the d i rec t p roduc t of p copies of G (r-i+1),

_ • ~ e j + 1 G ( r - i + 1,p) may be wr i t t en as the p roduc t ~I~P~:~ " " Yi,~+t, where ~ e . Therefore, if the

order of G (r-~+l'p) is m, the order of G (r-id+l) c a n n o t exceed m.p j÷l. However, recal l ing

the def ini t ion of the maps h i, the order of G (r-id+l) is a t l eas t m.p j+l. Therefore, the

factor group G ( r - i j + l ) / G (r-id) has o rder p.

Finally, G (r-id+1) is a n o r m a l subgroup of P, since the factor group G(r-i'J+l)/G (r-id)

iies in the center of P/G (r-id).

L ~ 18

Let ~I ..... ~r be permutations of order p determining a p-step central series for PI, If

G (r-i'p) is a n o r m a l subgroup of P and is the d i rec t p r oduc t of p copies of G (r-i), t h e n

G(r-i- l '~)/G (r-~'p) is in the c e n t e r of. P / G (r-i'p) and is of o rder p. F u r t h e r m o r e , G (r- i-m)

is also a n o r m a l subgroup of P.

Proof Clearly ~ + ~ i = hi(~+1) c o m m u t e s with X. I t also c o m m u t e s with every

~0 ~ H modulo G (r-i'p), s ince G(r-i-1)/G (r-i) is in the c e n t e r of P1/G(r-i), and since G (r-i'p)

is the d i rec t p roduc t of p copies of G (r-i). Thus, G(r - i - I ' l ) /G (r-i'p) is in the cen t e r of

P / G (r-i'p), and f rom this follows that G (r-i- l ' l ) is n o r m a l in P.

Observe t h a t we adjoin to G (r-~,p) an e l e m e n t ~Pi+l,1 of o rder p. By the same argu-

m e n t as for Lemma 17, this shows t h a t the order of the fac tor group Gfr-~-~'~)/G (r-i'p)

is a t mos t p. Since ~Pi+l is no t in G (r-~), O (r-~'p) is a p rope r subgroup of G (r-i-l ' l), and so

the fac tor group is of o rder p. m

CoRoI~ '~ 6 (Hoffmann)

Given a group PI with a p-s tep c e n t r a l ser ies i nduced by ~Pl . . . . . ~r, p e r m u t a t i o n s of

order p, the sequence ~ ,~ . . . . . ~P~,~, ~ ,~ . . . . . ~Pr,p, X d e t e r m i n e s a p-s tep cen t r a l ser ies

for P = P~%Cp, where ~P~d = hj(~) . F u r t h e r m o r e , the p e r m u t a t i o n s in this new

sequence are alt of o rder p.

Proof We es tab l i sh the r e su l t by i nduc t ion on the sequence m e m b e r s . The base

case is covered by Lemma !5. The i nduc t i on step is covered by L e m m a t a 17 and la.

The a s s u m p t i o n t h a t G (r-~'p) is the p-fold d i r ec t p r oduc t of G (r-i) is d i scharged induc-

t ively by the fact t ha t all fac tor groups are of o rder p, and by the def ini t ion of the

maps h i. Finally, observe t h a t C(0,:) is the subgroup H of P = P~%Cp and thus has

index p in P. F u r t h e r m o r e , <G(°'P),X> = P and P / C (°'p) is Abelian (cf. Theorem 8).

Therefore, adding X as the las t e l e m e n t comple te s the c o n s t r u c t i o n of a p-s tep c e n t r a l

series. Note t h a t X is a p e r m u t a t i o n of o rder p.

105

We a l r e a d y o b s e r v e d t h a t the m a p s hj p r e s e r v e the o r d e r of the p e r m u t a t i o n s

mapped . Therefore , the s e q u e n c e c o n s t r u c t e d also cons i s t s of p e r m u t a t i o n s of o r d e r

p. •

As a consequence of L e m m a t a 12, 13, and Corol la ry 6, we now have a r ecu r s ive

Solution t o P r o b l e m 5. We i l l u s t r a t e the c o n s t r u c t i o n with two example s .

~ I ~ 8

Let P be a Sylow p - subg roup of $25, where p = 5. Thus P is i somorph i c to C5%C5.

Assume t h a t P has t he se t s of i m p r i m i t i v i t y t i ..... 51, I6 ..... 10t . . . . . [2t ..... 25]. We ob t a in

the following vec to r s for the exponen t s in the maps hi, which m a y be r e d u c e d modulo

5:

hi:

hs:

h3:

54:

hs:

(1,1, t , l , l )

(o,1,~,3,4)

(o,o,1,~,6) = (o ,o , t ,3 ,1)

(o,o,o,1,4)

(o,o,o,o,1)

Note t h a t the e x p o n e n t vec to r s , be fo re r e d u c i n g modu lo 5, con ta in d iagona l co lumns

of P a s c a l ' s t r i ang le .

Using the m e t h o d of Corol lary 6, we ob ta in the following e l e m e n t s of P inducing a

p - s t e p c e n t r a l se r ies , where 7r = (1,2,3,4,5):

hl('n') = ( ' r r l ,~ ' ,~ ] ,~ ,~ l ; O) = (1,2,3,4,5)(6,7,8,9,10)._(21,22,23,24,25)

hz(~t) = (n°,~rl,n~,Tr~,n4; 0) = (6,7,8,9,10)(11,13,15,12, t4)_.(21,25,24,23,22)

h3(~T) = (n°,~°,~l,~r~,~l; 0) = (i1,12,13,14,15)(16, t9,17,20,18)(21,22,23,24,25)

h4(vt) = (~0,~0,~0,~,~4; 0 ) = (16,17,18,19,20)(21,25,24,23,22)

h~,(n) = (n°,-rr°,n°,'n'°,nl; O) = (21,22,23,24,25)

) / = (0,0,0,() ,0;~T) = (1,6,11, i6 ,2i)(2,7,12, i7,22).. .(5,10, i5,20,25),

Note t h a t each of the p e r m u t a t i o n s has o r d e r 5. In the sequence wr i t ten , t he p e r m u -

t a t ions d e t e r m i n e a c e n t r a l s e r i e s for P with quo t i en t s izes equal to 5. [:]

EXaMP~ 9

As an e x a m p l e for i t e r a t i n g our cons t ruc t i on , we cons ide r t he case p=3, and give the

g e n e r a t o r s for the c e n t r a l s~r ies of Sylow 3-subgroups of the s y m m e t r i c g roups of

deg ree 3, 32, and 33 . Let X = ( 0 , 0 , 0 ; ~ ) , 7r = (1,2,3). F o r deg ree n=3, we ob t a in the

s equence ~. For d e g r e e n=9, we ob ta in

hl(Tr ), hz(Tr), h3(n), X.

Finally, for n=27, we ob ta in

!06

hi(hi(tO), h21h1[~T;j,' '~' h~(h~(~)),.

hl(hs(~)) , h2(h3(~)), hs(h3(~)),

hi(x), h2(x), hs(x),

where e is ( 0 , 0 , 0 ; 0 ) , the i d e n t i t y in Cs%C 3.

Having shown how to c o n s t r u c t a c e n t r a l s e r i e s for P, a Sylow p - s u b g r o u p of t he

symmetric group, we next discuss the time required to determine the generators for

the ser ies .

It is c lea r t h a t the exponen t s for the m a p s hj can be d e t e r m i n e d in O(p 2) s teps .

F u r t h e r m o r e , if ~P is a p e r m u t a t i o n of deg ree m, t hen hj(~) is a p e r m u t a t i o n of deg ree

p.m, and can be c o n s t r u c t e d in O(p.m) s teps . To see this , obse rve t h a t in O(p.m)

s t eps we may c o m p u t e the f irst p powers of ~P. Having t h e s e avai lable , t he p e r m u t a -

t ion hi(~) can t h e n be c o n s t r u c t e d wi thin t he s a m e t ime bound.

n-__LI L e t P be a S y l o w p - s u b g r o u p of S n, w h e r e n = p k . T h e n P has o r d e r pP-1 = p p - I

It follows that we have to construct O(n__) permutations to determine a central series P

for P. Each permutation is obtained by applying the maps h i exactly k-s times to a

permutation of degree pS hence each can be constructed in O(n) steps. Thus we can

find a central series for P in O(~-~+p 2) steps. Observing that a~+b2- < (a+b) ~, where

a, b -~ 0, and that p ~ n, we obtain

THEORgM 16 (Hoffmann)

Let P be a Sylow p-subgroup of S n for which we have a r e c u r s i v e d e c o m p o s i t i o n into

se t s of i m p r i m i t i v i t y of s izes p, p2, p 3 e tc . Then g e n e r a t o r s for a c e n t r a l s e r i e s of P

m a y be d e t e r m i n e d in O(n 2) s t eps .

We conclude by showing how to m a k e every p-group H po lynomia l ly acces s ib l e

from a containing Sylow p-subgroup of Sn. We need here the following

LEMI~A 19

Let A, B be subgroups of G. If A is a normal subgroup of G, then the complex

AB = i c~ I c¢ E A, ~ c B ! is also a subgroup of G.

Proof Since A is normal in G, AD = BA, from which the lemma follows. -

107

We will now t r a p H < P. Le t G (0, r m i-~ 0, be t he g r o u p s in a c e n t r a l s e r i e s for P.

C o n s i d e r t h e fol lowing s u b g r o u p t o w e r of P:

I = G(r )~H <~ • • • '~ G(°)f~H = H = HG (r) < • ' • < HG (°) = P

This t o w e r c l e a r l y t r a p s H. We will show t h a t t h e i n d e x of s u c c e s s i v e q u o t i e n t s d o e s

n o t e x c e e d p.

By L e r n m a 9 of C h a p t e r II, t h e i n d e x of G(i+I)c~H in G(i)c~H is n o t l a r g e r t h a n t h e

i ndex of G (i÷I) in G (i), and is t h e r e f o r e n o t l a r g e r t h a n p. More p r e c i s e l y , s ince p i s a

p r i m e , e i t h e r G(i+I)NH is e q u a l to G(i)(~I-l, or i t has i n d e x p in t h a t g roup . Note t h a t

G(i+I)f~H is normal in G(i)c~,H.

For determining the quotient sizes in the upper portion of the subgroup tower we

need

L~A 20

LetA, B be subgroups of G. Then the order of AN is I ABI = I AI'IB[ IAnBI "

Proof In general AB is not a group, but it must contain complete right eosets of

A. We will put the right cosets of A contained in A}3 into I-i correspondence with the

right cosets of C = Af~B in B:

Let An and A~ be distinct right eosets of A in AB, where we assume, without loss of

generality, that n and ~ are in B. Then 7r~ -t ¢ A. Since 7r, 9 ~ B, CTr and C9 must be

distinct right cosets of C in B. Conversely, if C~ and C~ are distinct right cosets of C

in B, then 7r~ -I is not in C. Since 7T¢ -I C }3, we have ~-I ~t A. Thus the number of

right cosets of C contained in B is equal to the number of right eosets of A contained

in AB, f r o m which t h e l e m m a follows. -

Us ing L e m m a 20, we now s e e that q u o t i e n t s in the u p p e r part of t h e t o w e r a r e of

sma l l index : We h a v e

(HG(0:HG0+I)) = , I.HG(i) I IHC~G0+I) I IHG(i+I)I = p- IHf~G(i)I = p-q

Here q is either I or 1 and thus the index of HG 0÷I) in HG 0) is either i or p. P

Note that we have generators for the groups G 0) and for H. By the results of

Chapter If, we can therefore test membership in the groups HG 0) in polynomial time.

Membership in Hf~G (i) is tested by testing separately membership in H and in G 0).

Now i t is eas i ly ve r i f i ed t h a t H is ( 2 , e ' p ) - a c c e s s i b l e fo r s o m e c o n s t a n t c.

108

3.5. Setwise S tab i l i ze rs i n p-Groups (Method i )

We now show how to efficiently d e t e r m i n e setwise s tabi l izers in a r b i t r a r y p-

groups. The a lgor i thm to be p r e s e n t e d makes use of the t echn iques developed in Sec-

t ions 3.3 and 3.4, and may be applied to devise a polynomial t ime i somorph i sm t e s t

for t r iva ten t graphs , as will be d iscussed in the n e x t chap te r .

We begin by cons ider ing how to d e t e r m i n e the setwise s tabi l izer in a Sylow p-

subgroup of the s y m m e t r i c group. The following is s t ra ightforward:

T,~:MMA 21

Let G < Sym(X) be the d i rec t p r o d u c t of the groups A < Sym(Xl) and B < Sym(X2), Y a

subse t of X. Then G¥ = Ay~xl×BTV~x ~,

As a consequence of the l emma, we only need to cons ide r the t r ans i t ive ease.

Thus, we will cons ider

PROI~.~ 6

Given gene ra to r s for a Sylow p-subgroup P of the s y m m e t r i c group of degree ph

h > 0, and a s u b s e t Y of the p e r m u t a t i o n domain, d e t e r m i n e gene ra to r s for Py, the

setwise s tab i l izer of Y in P.

The idea for solving P r o b l e m 6 is as follows: We cons ider the cone g raph X = (V,E)

associa ted with P and label each leaf with one of two labels according to whe ther the

cor responding poin t in the p e r m u t a t i o n domain of P is in X. We t h e n d e t e r m i n e gen-

e ra to rs for t h e subgroup of a u t o m o r p h i s m s of the graph which r e s p e c t this labelling.

All s teps are s t ra ightforward, except the last, which we accompl ish using a va r i a n t of

the ~ree isomorphism ~lgori~hrn.

We s u m m a r i z e the t ree i somorph i sm algor i thm, and discuss how to modify it for

our purposes . Let V k be the se t of ver t ices in the t r ee (we wilt cons ider the BFS-tree

of X) which are a t d i s tance k f rom the root. P roceed ing from the leaves to the root,

we cons ider each se t V k, and classify the sub t r ee s roo ted in the ver t i ces in Vk in to iso-

rnorphisrn classes. To each ve r t ex v in V k we a t t a c h a n u m b e r ident i fying the i somor-

ph ism class of the sub t r ee rooted in v. It is c lear how to do this classif icat ion for the

label led leaves. Let v be in V k with i ts r sons in Vk+ 1 having b e e n label led i I . . . . . i r. We

assign to v the r - tup le (i I, . . , it). Now le t w E V k be a no t he r ve r tex to which we have

109

ass igned the t up l e 01 . . . . . Jr). Then v and w belong to t he s a m e i s o m o r p h i s m c lass iff

t h e r e is a p e r m u t a t i o n of t h e t up l e a s s igned to v which is equa l to t he t up l e a s s igned

to w. All p e r m u t a t i o n s a r e allowed, s ince the s u b t r e e s of v m a y be p e r m u t e d f ree ly .

Consequent ly , we m a y ass ign the t up l e s so t h a t the c o m p o n e n t s a re in s o r t e d order .

Having a s s igned the ( sor ted) tup les , the d i s t i n c t occu r r ing t up l e s a r e e n u m e r a t e d and

the r e su l t ing n u m b e r of a t up l e is a s s igned to eve ry v e r t e x l abe l l ed with t h a t tuple .

The a s s igned n u m b e r s se rve as labels of the i s o m o r p h i s m c lasses of t h e s u b t r e e s .

By a jud ic ious choice of d a t a s t r u c t u r e s and su i tab le sor t ing m e t h o d s for t he

tup les c o n s t r u c t e d , the above a l g o r i t h m m a y be i m p l e m e n t e d to run in t ime p r o p o r -

t ional to t he n u m b e r of ve r t i ces , i r r e s p e c t i v e of the m a x i m u m n u m b e r of sons of any

t r e e ve r tex .

Now c o n s i d e r ou r case where X is the r e g u l a r cone g r a p h a s s o c i a t e d with t h e

group P. Since the n o n t r e e edges have to be p r e s e r v e d , we m a y only p e r m u t e the

s u b t r e e s of a v e r t e x v cycl ical ly . Thus i t would be i n a p p r o p r i a t e to so r t t he tup le

c o m p o n e n t s . Ins tead , le t (i i . . . . . ip) be t he t up l e of l abe ls a t t a c h e d to the sons of v in

the cycl ic o rde r ing of t he sons (i.e. following the n o n t r e e edges connec t ing t h e sons).

There a r e p poss ib le a r r a n g e m e n t s of the t up l e c o m p o n e n t s , t hus up to p d i s t i n c t

tup les a re poss ib le . We will ass ign to v the one which is l ex i cog raph iea l ly first .

Modified in th is way, i t is c l ea r t h a t we c o r r e c t l y c lass i fy t he s u b t r e e s of X along with

i n c i d e n t n o n t r e e edges in to i s o m o r p h i s m c lasses . This modi f i ca t ion i n t r o d u c e s t he

f ac to r p in to t he runn ing t i m e bound.

Let p=5, and a s s u m e v is t he v e r t e x shown in F igure 19 below, occu r r ing in t he cone

g r a p h X of d e g r e e 5 a s s o c i a t e d with some Sylow 5-subgroup of a s y m m e t r i c g roup of

d e g r e e 5 h. Let 1,1,2,2, and 3 be t he i s o m o r p h i s m c la s ses of t he sons of v, as shown.

1 2

v

(z,~,2,t.3) Figure 19

110

We may assign to v any one of the five tuples (i,3,1,2,3), (2, i,3,!,2), (&2,I,3,!),

(1,2,3,1,3), (3, I,2,2,1). Here, the tuple (1,2,2,1,3) is lexicographicaily first and is

assigned to v. []

Having classified the s u b t r e e s in to i somorph i sm classes, t he re is no difficulty

obta ining gene ra to r s for the a u t o m o r p h i s m group. We associate with each ver tex

v ~ V k a p e r m u t a t i o n ~r v which exchanges the sub t r ee rooted in v with the sub t r e e

rooted in w, where w ~ Vk is an a rb i t r a r i l y chosen r ep re sen t a t i ve in the i somorph i sm

class of v. It is c lear how to c o n s t r u c t these p e r m u t a t i o n s in a single pass f rom the

leaves to the root. F rom them, it is obvious how to ob ta in the genera to r s , and the

r eade r should have no difficulty in working out the detai ls and proving

T H E O ~ ~7 (Hoffmann)

Let P be a Sylow F-subgroup of S m n = ph X = (V,E) the assoc ia ted d i rec ted regu la r

cone graph of degree p and he ight h. If Y is any s u b s e t of ~1 ..... nl , t h e n gene ra to r s for

Py can be d e t e r m i n e d in 0(r,&p) steps. F u r t h e r m o r e , the r e su l t ing gene ra t ing se t is

of size O(n).

Thus we have a po lynomia l t ime solut ion for P r ob l e m 8. Because of Lemma 21,

the bound of the t h e o r e m also applies to Sylow p-subgroups P of Sn where n is no t a

power of p. The bound of _Theorem 17 m a y be lowered to O(n-p) if the gene ra to r s are

represented by a special data structure (see Section 4).

Finally, we give the algorithm for computing the setwise stabilizer in a p-group.

111

ALC~ORITHM 3 (Setwise Stabilizer in a p-Group)

Input

Output

Method

1.

A genera t ing se t K for the p-group G of degree n, and a subse t Y of

A genera t ing se t K' for G¥, the setwise stabil izer of Y in G.

Construct a Sylow p-subgroup of S n containing G as subgroup, and construct the

associated collection of directed regular cone graphs of degree p, whose auto-

morphism group is isomorphic to P.

2. Decompose P into the direct product of its transitive constituents, PI ..... Ps,

where Pj has degree a power of p.

3. For i <- j -< s, let Yj be the intersection of ¥ with the permutation domain of Pj.

Using the corresponding cone graph, construct a generating set Kj for the set-

wise stabil izer of Yj in Pj. Note t h a t P¥ = <K 1 ... . . Ks>.

4. Cons t ruc t a cen t ra l series for P.

5. Using the cen t ra l series, make G (k,c)-accessible f rom P by t rapping it in a sub-

group tower of P.

6. I n t e r s ec t the tower t rapping G with Py, a subgroup of P with known genera tors ,

t he reby de termining gene ra to r s for C~.

We analyze Algori thm 3:

Steps 1 and 2 are done using the techniques of Sect ion 3.3, and require, by Corol-

lary 6, O(IKt-n&log2*(n)) s teps. The derived genera t ing se t for P contains at m o s t n

permuta t ions .

Step 3 is done using the modified t r ee i somorphism algori thm, and takes O(n&p)

s teps (Theorem 17).

Step 4 is accompl i shed using the techniques of Sect ion 3.4. It requires 0(n 2)

s teps (Theorem 16).

For Step 5, we m ay use the techniques of Chapter ti to derive an O(n 2) m e m b e r -

ship t e s t in each group. As the re m a y be up t o n groups to consider, we need here

0( lKI .n2+n 7) steps, observing that , excep t for K, we have small genera t ing sets for

each group.

1/2

For Step 7, we use Algorithms 6 and ? of Chapter II, applying Theorem 14 of

Chapter If. By Proposition 5 of that chapter, this step requires O(nS.p s) steps.

In summary, we have proved

TIIEOR~ 18 (Hoffmann)

Let G = <K> be a p-group of degree n, Y a subse t of the p e r m u t a t i o n domain. Then

gene ra to r s for C~, the setwise s tabi l izer in Y of G, can be d e t e r m i n e d in

0(IKI.nZ-iogz*(n)+p3.nS+n 7) s teps,

Thus, Algori thm 3 is a polynomial t ime a lgor i thm for finding the setwise s tabi l izer

in a p-group.

4. Notes and References

Sect ion 1 is based mos t ly on Babai [ 1979], who first proposed P rob lem I and gave

a r a n d o m polynomial t ime a lgor i thm for it. Furs[ , Hopcroft and Luks [1980a]

discovered a de t e rmin i s t i c solut ion for P rob lem f. Babai 's me thod for genera t ing uni-

formly d i s t r i bu t ed r a n d o m e l e m e n t s in a p e r m u t a t i o n group is d i f ferent and applies

only to symmetric groups. The method given here (Algorithm I) is apparently new.

Cone graphs were first considered by Hoffmann [1980a]. The original class

definition given in that paper differs from the one given here (Definition i) :in that the

BFS-tree was required to be balanced. Our exposition of the material in Section 2 also

differs in other respects: regular cone graphs (Definition 2) are called seuziregz~=r in

Hoffmann [1980a]. Furthermore, the indexing of the groups A C~) has been changed to

make it consistent with the indexing in other subgroup towers. Hoffmann [1980a]

gave a probabilistie 0(n c'1°g2(n)) isomorphism test for regular cone graphs, using the

probabi t i s t ic a lgor i thm of Babai 's , Because of the inappl icabi l i ty of L e m m a t a 2 and 3

to the groups A, (k), k > 1, the a lgor i thm is incor rec t , One may c o r r e c t i t in the b ina ry

case using some of the ideas of Sect ion 3 Of this chapter . However, one can go f u r t he r

and ob ta in a polynomial t ime a lgor i thm for this class dropping the k - i somorph i sm

approach. We will descr ibe this me thod in Chapter IV.

Most i n t r o d u c t o r y texts on Group Theory will con ta in a thorough t r e a t m e n t of the

e l e m e n t a r y p roper t i e s of p-groups and Sylow p-subgroups , as well as of Theorems l l

and 12. Our exposi t ion of this m a t e r i a l by and large follows Kochend~rffer [ 1970] and

Hall [ 1959].

113

The construction of the Sylow p-subgroups of the symmetric group in terms of

direct products and wreath products is due to Kaloujnine [1948]. By now the con-

struction is standard material and is given in most texts on Group Theory. While

there seems to be no explicit mention in the literature, the relationship between

these p-groups and cone graphs is implicitly well-known to mathematicians. Furst,

Hopcroft, and Luks [1980a] were first to explicitly exploit this relationship for the

purpose of testing isomorphism of trivalent graphs (see also Chapter IV). In particu-

lar, they considered the case p=2, for which they gave an algorithm for constructing a

Sylow 2-subgroup of the symmetric group containing a given r-group. They also

derived the central series for Sylow 2-subgroups of S n and showed how to use it to

make any r-group polynomially accessible.

Algorithm 2 for finding a minimal set of imprimitivity containing a prescribed

pair of points is originally due to Atkinson [1975]. Atkinson's algorithm requires

0(IKI.n 2) steps for finding the set of imprimitivity. Thus, determining whether <K> is

imprimitive would require O(IKI 'n s) steps. Atkinson discusses some improvements

which lower this bound to 0(IKI.n~.log2(n)). Surprisingly, the application of the dis-

joint set union/find algorithm is new. Once the applicability is noticed, it is easy to

produce Algorithm 2 from Atkinson's method thus further lowering the bound to

O( IKI "nZ'Ioga*(n)). In fact, there is a striking similarity between Algorithm ~ and the

first order unification algorithm of Baxter [ 1976], and the data structures are nearly

identical. For an analysis of the fast disjoint set union/find algorithm see Aho, Hop-

croft and Ullman [ 1974].

The generalization of the construction of a central series to arbitrary primes in

terms of the maps h i seems to be new. The standard method for constructing p-step

central series in p-groups is to form commutator subgroups. While this approach is

formally very elegant, it does not lead to a more efficient method for finding a series

in the Sylow p-subgroups of S n. We will give an algorithm for constructing the commu-

tator subgroup series of a permutaiion group in Chapter VI.

The application of trapping the p-group G to setwise stabilizers in p-groups, and

the ensuing polynomial time isomorphism test for %rivalent graphs (see also

Chapter IV) seems to be new, although the techniques are implicit in Furst, Hopcroft,

and Luks [1980a]. The tree isomorphisms algorithm is described and analyzed in Aho,

Hopcrof t , and Ul lman [1974]. An a l g o r i t h m for d e t e r m i n i n g the a u t o m o r p h i s m group

of t r e e s in l i nea r t ime m a y be found in Colbourn and Booth [1980].