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K. K. Agarwal [1978]

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Structures w~th Application to a Heuristic Program for the S y n t h e ~ of

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A. V. Aho, J. E. Hopcroft, J. D. Ullman [ 1974]

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V. L. Arlazarov, I. I. Zuev, A. A. Uskov, t. A. Faradzhev [1976]

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M. D, Atkinson [1975]

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S. Baase [ 1978]

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L. Babai [i980] "On the Complexity of Canonical Labelling of Strongly Regular Graphs";

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L. Babai, P, grdSs [1978]

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L. Babai, L. Kucera [1979]

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L. Babai, L. Lovasz [i973]

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T. Beyer, A. Proskurowski [ 1978]

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R. J. Bonneau [1973]

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K. S. Booth, C. S. Lueker[ 1975]

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255 - ~65

276

H. Brown, R. Bueiow, J. Neubiiser, H. Wondratschek, H. Zassenhaus [1978]

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J. J. Cannon [ 1973]

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J. J. Cannon [ 1974]

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R. E. Carhart [ 1978]

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L. Car ter [1977]

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C. J. Colbourn, K. S. Booth [1980]

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M. J. Colbourn, C. J. Colbourn [1978]

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M. J. Colbourn, C. J. Colbourn [1979]

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S. A. Cook [1971]

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D. G. Corneil [1972]

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D. G. Corneil [1974]

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A. K. Dewdney [1978]

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A. Dietze, M. Schaps [1974]

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P. Dreyer [1970]

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L. E, Druffet [1975]

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L. E. Druffel, D. C. Schmidt, J. E. Simpson [1975]

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Proc. lOth ACM Symp. on Thy. of Comp., (1978) 133- i42

I. 5. Filotti, J. N. Mayer [1980]

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I. 5. Filotti, G, L. Miller, J. Reif [1979]

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M. Fontet [ 1974]

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C. M. Hoffmann ~ [ 1980aj

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(1976) 3 - 8. Edited by R. D. Jenks

1~I. Overton, A. Proskurowski [1975]

"Findhug the Maximal Incidence Matrix of a Large Graph"; Tech. Rept. STAAV-

CS-75-509, Stanford Univ. (1975)

A. Ja. Petrenjuk [1975]

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V. Pless [1975]

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2 I4 - 220

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A. Proskurowski [1973]

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M. Randic [1974]

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M. Randic [1975]

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M. Randic [1977]

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R. C. Read [1972]

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R. C. Read, D. G. Corneil [i977]

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R, Remak [1930]

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H. Robertz [i976]

"Eine Me~hode zur Berechnung tier Automor~phisrnengruppe ei, ner Endgi-

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292

J. B. Rosser, L. Schoenfeid [i975]

"Sharper Bounds for the Chebyshev Functions @(x) and ~I,(x)"; Math. of

Comp. 29, (i975) 243 - 269

D. C. Schroidt, L. E. Druffel [ i976]

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Edited by G. Birkhoff and M. Hall, Jr.

C. C. Sims [ i97ib]

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Algebraic Manipulation, Los Angeles, Calif., ( i97i) 23 - 28. Edited by S. R.

P etr ick

C. C, Sims [1978a]

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C. C. Sims [1978b]

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C. C. Sims [1978c]

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INDICES

1. Problem Index

Chapter II:

Problem

Problem

Problem

Problem

Problem

Problem

Problem

Problem

Chapter IIi:

Problem

Problem

Problem

Problem

Problem

Problem

Chapter IV:

Problem

Problem

Problem

Chapter V:

Problem

Problem

Problem

Chapter Vh

Problem

Problem

Problem

i: Existence of Graph Isomorphism

2: Existence of Isomorphism of Labelled Graphs

3: Graph Isomorphism

4: Graph Automorphism

5: Order of the Automorphism Croup

6: Number of Isomorphisms

7: Group Specification by Generators

8: Determining an Accessible Group

i: Labelled Graph Automorphism

2: Automorphism of Cone Graphs of Fixed Degree

3: Sylow p-Subgroup of Sn Containing a Given p-Group

4: Minimal System of Imprimitivity for p-Groups

!nduced Factor Groups

5: p-Step Central Series for Sylow p-Subgroup of S n

5: Setwise Stabilizer in p-Groups

i: Trivalent Graph Aufiomorphism

2: Automorphism for Binary Cone Graphs

3: Imprimitivity Problem for 2-Groups

i: Automorphism of Graphs of Fixed Valence

8: Setwise Stabilizer in Groups in Fb

3: Sylow p-Subgroup of a Primitive Croup in F b

i: Double Coset Membership

2: Croup Facfiorization

3: Number of Factorizations

and

24

24

24

25

25

25

33

5i

6i

76

91

94

100

i08

1118

i16

i58

179

1188

215

236

28,6

236

297

Chapter VI:

Problem 4:

Problem 5:

Problem 6:

Problem 7:

Problem 8:

Problem 9:

Problem i0:

Problem i i :

Problem i2:

Pi 'oblem i3:

Problem i4:

Problem i5:

Problem i6:

Problem i7:

Problem i8:

Problem i9:

Problem 20:

Problem 2i:

Problem 22:

Problem 23:

Problem 24:

Problem 25:

Problem 26:

Problem 27:

Coset In tersec t ion Emptiness

Group In tersec t ion

Setwise Stabil izer

Centralizer in Another Croup

Rest r ic ted Graph Automorphism

Existence of Graph Isomorphism

Graph Automorphism

Small Graph with Prescr ibed Automorphism Croup

In tersec t ion o~ Commuting Groups

In tersec t ion with a Normalizing Croup

In tersec t ion with a p-Croup

Intersect ion with a Croup in F b

In tersec t ion of two Accessible Subgroups

In tersec t ion with an Accessible Subgroup

Centralizer in S n

Center

Intersection of the Automorphism Group of a Graph with

a Sylow p-Subgroup of Sym(V)

Normal Closure

Commutator Croup

Solvability

Nilpotenee

Double Coset Par t i t ion

Intransi t ive Subgroup Problem

Subspaee Problem

238

238

239

24i

24i

245

245

246

246

248

248

248

248

248

252

252

260

262

268

268

268

270

270

27i

298

2. Algori thm Index

Chapter II:

Algorithm ~:

Algorithm 2:

Algorithm 3:

Algorithm 4:

Algorithm 5:

Algorithm 6:

Algorithm 7:

Chapter III:

Algorithm i:

Algorithm 2:

Algorithm 3:

Chapter IV:

Membership Test in a Pe rmuta t ion Group G

Sift

Construct ion of a Representa t ion Matrix from Genera-

tors

Point Orbit and Schreier Vectors

Strong Generating Set for <K> Relative to [i ..... n]

Generalized Sift

Determination of a Subgroup Tower

Probabil is t ic Method for Problem I

Set of Imprimit iv i ty

Setwise Stabilizer in a p-Group (Method I)

Algorithm 1: Automorphism of a Connected, Trivalent Graph

Algorithm 2:

Algorithm 3:

Chapter V:

Algorithm i:

Algorithm 2:

Algorithm 3:

Chapter VI:

Algorithm 1:

Algorithm 2:

Algorithm 3:

Setwise Stabi l izer in a p-Group (Method 2)

In tersec t ion of two p-Groups

Automorphism of Graphs of Valence d

Subgroup with p-Group Action

Setwise Stabil izer in Fb

In te rsec t ion with a Normalizing Group

Central izer in S n

Normal Closure

34

38

39

46

48

53

54

7i

96

Iii

126

I34

153

182

220

250

259

264

3. Def in i t ion Index

299

Chapter II:

Definition 1:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Definition 9:

Definition 10:

Definition 11:

Definition 12:

Definition 13:

Definition 14:

Definition 15:

Definition 16:

Definition 17:

Definition 18:

Definition 19:

Definition 20:

Definition 21:

Definition 22:

Chapter tII:

Definition I:

Definition ~:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Group

Subgroup

Left Cosets, Right Cosets

Normal Subgroup

Group Homomorphism

Conjugates

Generating Set

Orbit in a Permutation Group, Orbit Length

Point Stabilizer

Pointwise Stabil izer

Setwise Stabi l izer

Simple, Undi rec ted Graph

Automorph i sm of a Graph

I somorphism ol two Graphs

Complemen t Graph

Polynomial Time Equivalence of Prob lems in NP

Vertex Labelled Graph

Represen ta t ion Matrix

Base of a P e r m u t a t i o n Group

Basic Orbit of a P e r m u t a t i o n Group

Strong Genera t ing Set

(k,c)-Accessibility of Groups

Cone Graph, Root of a Cone Graph

Regular Cone Graph

Degree of a Cone Graph

k-Automorphism of a Cone Graph

k-lsomorphism of Cone Graphs

Simple Extension

p-Group

Sylow p-Subgroup

12

14

14

15

16

t6

18

18

i9

ig

19

21

21

22

23

24

24

35

44

44

47

55

73

73

74

77

78

83

85

86

300

Chapter !II:

Definition 9:

Definition i0:

Definition i i:

Chapter IV:

Definition i:

Definition 2:

Definition 3:

Chapter V:

Definition I:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Definition 9:

Definition i0:

Definition i i:

Definition !2:

Definition i3:

Definition i4:

Definition 15:

Definition 16:

Definition 17:

Definition 18:

Chapter VI:

Definition i:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Center of a Group

Central Series, p-Step Central Series

Wreath Product

Commutator

Composition Series, Imprimitivity series

Composition Sequence, Imprimitivity Sequence

Simple Group

Subnormal Series

Section

Maximal System of Imprimitivity

Gi-component of Subgroups of GIxG 2

Centralizer in Another Group

Norrnalizer in Another Group

Right and Left Regular Representation

Semiregular Group

Characteristic Subgroup

Characteristically Simple Group

Minimal Normal Subgroup

Socle

E l e m e n t a r y Abelian Group

Genera l Linear Group

Full Diagona! Subgroup

Order of a Modulo p

Forward Connect iv i ty in Graphs

Double Coset

Cycle Graph of a Permutation

Labelled Cycle Graph of a Set of Permutations

Cycle Distance

Normal Closure

Commutator

86

87

88

140

143

144

184

i84

186

!89

i94

i95

195

197

i97

199

2OO

20O

2OO

203

2O4

2O7

211

228

232

253

254

254

262

265

301

Chapter VI:

Definition

Definition

Definition

7: Commuta tor Group, Derived Group

8: Solvable Group

9: Nilpotent Group

265

266

267

4. L e m m a Index

302

Chapter ti:

L e m m a

L e m m a

L e m m a

Lemma

L e m m a

L e m m a

L e m m a

Lemma

L e m m a 9:

Chapter IIh

L e m m a 1:

L e m m a 2:

L e m m a 3:

L e m m a 4:

L e m m a 5:

L e m m a 6:

Lemma 7:

L e m m a 8:

L e m m a 9:

L e m m a 10:

L e m m a i i :

L e m m a 12:

L e m m a 13:

L e m m a t4:

L e m m a 15:

L e m m a 16:

!: Polynomial Time Reduction of Problem 2 to Problem 1

2: Polynomial Time Reduction of Problem 3 to Problem 2

3: Complete Right Transversals of a Point Stabilizer Sub-

group Tower Define Generating Sets


5: Polynomial. Time Reduction of Problem 5 to Problem 4

6: Polynomial Time Reduction of Problem i to Problem 5

7: Rows in a Representation Matrix are Right Transversals

8: Comple te Right Transversa ls in a Subgroup Tower Define

Genera t ing Sets

Inequal i ty of Indices

A Group from which Aut(X) is Accessible, X a Graph as in

Problem I

Bound on the Subgroup Indices above Aut(X)

Bound on the Subgroup Indices below Aut(X)

Generating Uniformly Distributed Random Integers

Random Sift

Probability of Obtaining a Complete Right Transversal by

Random Sift

Normality of A (k+1) in A (k)

A Group Containing Aut(X), X a Regular Cone Graph

Membership in A, (k)

Finiteness of Algorithm 2

Checking a System of Imprimitivity from a Generating

Set

p-Step Central Series for Cp

p-Step Central Series for a Direct Product

Subgroup Structure of A%B

A Central Subgroup of PI%Cp

CommutLng Properties of h i

25

26

27

28

29

29

36

,5!

56

64

64

65

67

69

69

78

82

84

96

97

i00

I00

101

10E

!03

303

Chapter III:

Lemma 17:

Lemma i8:

Lemma i9:

Lemma 20:

Lemma 21:

Chapter IV:

Lemma i:

Lemma 2:

Lemma 3:

Lemma 4:

Lemma 5:

Lemma 6:

Lemma 7:

Lemma 8:

Lemma 9:

Lemma 10:

Lemma 1 i:

Lemma i2:

Lemma i3:

Chapter V:

Lemma

ae m m a

Lemma

Lemma

Lemma

Lemma

Lemma

Lemma

Lemma

Properties of the Croups G (i'D

Properties of the Croups G (i'j)

If A normalizes B, then AB is a group

The order of AB

Setwise Stabilizers in an External Direct Product

Automorphisms Respect Family Types

Conditions for Extending a c Aute(Xk)

Generating Set for Aute(Xk+l)

Coset Intersection

Maximal Systems of Imprimitivity in p-Groups

Timing of Algorithm 2 for Transitive p-Croups

Schreier Generating Set for Subgroups

Elementary Properties of J(Afi,A)

Recursion Rules for J(A~,A)

PasUn~ Cosets J(A'~i~A )

On Point Stabilizers in the 3ocle of Primitive Croups

Imprimitivity Structure of Pair Action

Deletion of Redundant Generators from a Composition

Sequence

i: Composition Series Induced by Normal Silbgroup and

Factor Group

2: Characterization of F b

3: Closure Properties of F b

4: Composition Series of Symmetric Groups

5: Internal Direct Product

6: Inn(GlxCs) ~ Inn(G1)xInn(C~)

7: A Regular Croup is the Right (Left) Regular Representa-

tion of Another Group

8: On Centralizers of Minimal Normal Subgroups of Primi-

tive Groups

9: Normalizer of a Full Diagonal Subgroup

103

i04

106

107

108

i2i

122

123

i30

i3i

137

i42

148

i48

t52

i59

160

i69

i85

i86

i87

i87

i93

i96

i97

202

207

304

Chapter V:

Lemma i0:

Lemma i l :

Lemma 12:

Lemma I3:

Lemma i4:

Lemma i5:

Chap t e r VI:

Lemma l:

Lemma 2:

Lemma 3:

Lemma 4:

Lemma 5:

Lemma 6:

Lemma 7:

Lemma 8:

Lemma 9:

Lemma i0:

Lemma i i :

Lemma i2:

Lemma ].3:

Subgroup S t r u c t u r e of Direct P roduc t s of Isomorphic

Nonabel ian Simple Groups

On Point Stabilizers of the Socle

(i+pSy) m = l+pSu

Highest Power of p Dividing (pt)

(i+pSy) pt = i+pS+tu with Except ions

Number of Subprob tems when Dete rmin ing Setwise Sta-

bil izers in p-Groups

Cardinai i ty of A~B

Number of Fac tor iza t ions in AB

Polynomial Time Equivalence of Prob lems 4 and 2

The Centra l izer as I n t e r s ec t i on of Genera to r Centra l izers

Centra l izer of a P e r m u t a t i o n

Necessary and Sufficient Conditions that AB is a Group

Sufficient Condition that AB is a Group

Point Stabilizers Inherit the Normalizing Property

Automorphism Group of a Cycle Graph

Automorphism Group of a Labelled Cycle Graph

Properties of Commutators

Properties of Commutator Groups

Commutator Properties of Central Series

207

208

211

211

2 i2

224

332

237

239

242

242

247

247

249

253

254

265

265

267

5. Proposition Index

Chapter Ih

Proposition

Proposition

Proposition

Proposition

Proposition

Chapter III:

Proposition

Chapter IV:

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

Chapter V:

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

305

1: Timing of Algorithm 1

2: Correctness and Timing of Algorithms 2 and 3

3: Correctness and Timing of Algorithm 4



1: St ructure of the Wreath Product of Symmetr ic

Croups

1: Determining a Nontriviai Sys tem of Imprimitivity in a

p-Group

2: Determining a Complete Imprimitivity St ructure for

a p-Group

3: Determining an Imprimitivity Sequence f rom

Generators

4: Setwise Stabilizer in a p-Group in O(n 3)

5: Determining a Sylow 2-SubgroupPof Sym(L 2) Contain-

ing the Pair Action of the 3-Group G < Sym(L)

6: Determining Auto(X) in O(nS), X a Simple Cone Graph

l: Cardinality of Elements with a Fixed Gl-component

2: Inn(G) is Normal in Aut(G)

3: Inn(G) is Faithful on a Normal Subgroup N iff

CdN) = I

4: The Centralizer of a Regular Group G is Isomorphic to

G

5: St ructure of *GxG* and Point Stabilizers

6: Minimal Normal Subgroups are Characterist ically

Simple

7: The Socle is a Characterist ic Subgroup

34

40

45

55

55

89

141

143

144

157

163

167

195

196

196

198

198

200

201

306

Chapter V:

Proposition 8:

Proposition 9:

Proposition 10:

Proposition 11:

Proposition i2:

Proposition !3:

Proposition i4:

Proposition i5:

Chapter VI:

Proposition i:

Proposition 2:

Proposition 3:

Proposition 4:

Proposition 5:

Proposition 6:

Proposition 7:

Proposition 8:

Proposition 9:

Nontrivial Normal Subgroups of Primitive Groups are

Transitive

Order of GL(m,p)

Normal Structure of Direct Products of Isomorphic

Nonabe!ian Simple Croups

A Primitive Group Can Be Embedded into the Auto-

morphism Croup of its Nonabelian Soele

The Abelian Soele of a Primitive Group is Elementary

Abelian

identities for (i +pSy) ptm

Timing of Algorithm 2 for Groups in F b

Timing of Algorithm 3 for Groups in Fb

Graph Isomorphism is a Double Coset Problem

Timing of Algorithm 1

Testing whether B Normalizes A

Intersection with a p-Group

Timing of Algorithm

Determining the Center in O(IKI "nZ+n a)

Problem 20 is in P


Solvability Criterion

202

204

204

206

2i0

212

217

225

233

~5I

251

251

259

26O

261

264

266

307

6. T h e o r e m Index

C h a p t e r I I :

Theo rem 1:

Theo rem 2:

Theo rem 3:

T h e o r e m 4:

T h e o r e m 5:

Theo rem 6:

Theorem 7:

T h e o r e m 8:

Theo rem 9:

Theorem lO:

Theo rem l l :

Theo rem i2:

T h e o r e m iS:

Theo rem i4:

C h a p t e r III:

T h e o r e m 1:

T h e o r e m 2:

Theo rem 8:

Theo rem 4:

Theo rem 5:

T h e o r e m 6:

T h e o r e m 7:

T h e o r e m 8:

Theo rem 9:

Lag range ' s T h e o r e m

Cayley 's Theo rem

C o r r e s p o n d e n c e of Orbi t Poin ts wi th Right Cosets of

Po in t S tab i l i ze r

The I s o m o r p h i s m s of a Graph F o r m a Right Coset of t he

A u t o m o r p h i s m Group

A Bound on the Minimal N u m b e r of G e n e r a t o r s

Po lynomia l Time Equivalence of P r o b l e m s i t h r o u g h 6

The A u t o m o r p h i s m Group of a Graph is the I n t e r s e c t i o n

of two Known P e r m u t a t i o n Groups

Necessary and Sufficient Conditions for a Representa-

tion Matrix to Specify a Group

P r o b l e m 7 is in P

Exis tence of a Smal l S t rong Genera t ing Se t

N e c e s s a r y and Sufficient Condi t ions for the Right

Transversa l s of a Subgroup Tower to be Comple te

P r o b l e m 8 is in P for (k ,c ) -Access ib le Groups

I n t e r s e c t i o n of two (k ,e) -Access ib le Groups

I n t e r s e c t i o n of a (k ,c) -Access ib le Group with a P e r m u t a -

t ion Group with Known C e n e r a t o r s

De t e rmin i s t i c Solu t ion of P r o b l e m i

Gene ra t i on of Uniformly D i s t r i bu t ed Random P e r m u t a -

t ions in S n

Random Algorithm for Determining a Subgroup Tower

Timing of Algor i thm 1

C h a r a c t e r i z a t i o n of A(k)/A (k+])

Cauchy ' s T h e o r e m

Sylow's Theo rems

Every p-Group has a Nontr ivial Cen te r

Every p-Group has a p -S tep Cent ra l Ser ies

14

i ?

19

22

25

29

3i

35

40

49

52

56

56

57

65

67

70

7 i

80

86

86

87

87

308

Chapter I I I :

Theorem lO:

Theorem l l :

Theorem t~:

Theorem l3:

Theorem l4:

Theorem l5:

Theorem l6:

Theorem I7:

Theorem l8:

Chapter IV:

Theorem i:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 6:

Theorem 7:

Theorem 8:

Theorem 9:

Chap te r V:

Theorem i:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 8:

Struct-~e of Sylow p-Subgroups of S n

Characterization of Primitivity and Imprimitivity

The Imprimitivity Sets of p-Groups

Loop Invariant for Algorithm 2

Correctness of Algorithm


Determining a p-Step Central Series of Sylow p-

Subgroups of Sn

Determining Setwise Stabilizers in Sylow p-Subgroups of

Sn

Correctness and Timing of Algorithm 3

If X is Connected and Trivalent, then Aura(X) is a 2-

Group

Timing of Automorphism of Connected, Trivalent Graphs

Relative to Setwise Stabihzation in 2-Groups

Determining Setwise Stabilizers in p-Groups

Lifting the Stabilizer of a Maximal Set of Imprimitivity

Determining J((GxH)K,X) in O(n 3)

Solution of the Imprimitivity Problem for 2-Groups in

O(nD Gadget Reduction for Extended Simple Cone Graphs

Determining the Elements of Ae(Xk) which may be

Extended to an Automorphism of Xk+ 1

Timing of the Iterative Step in Comput ing Ae(X )

Timing of Problem i Relative to Setwise Stabilization

Jordan-HiJlder' s Theorem

Composition Series of Subgroups

AULe(X ) E Ff(d), where X is Connected of Valence d

The Centralizer of a Transitive Group is Semiregular

Conjugation of a Regular Normal Subgroup Under Ele-

ments of a Point Stabilizer

89

92

92

97

98

98

i06

l lO

i12

l t 5

128

138

145

157

i65

169

!74

i75

184

i85

185

187

t98

!99

309

C h a p t e r V:

T h e o r e m 7:

T h e o r e m 8:

Theo rem 9:

Theo rem 10:

C h a p t e r VI:

T h e o r e m 1:

Theo rem 2:

T h e o r e m 3:

Theo rem 4:

Theo rem 5:

Theo rem 6:

T h e o r e m 7:

Theo rem 8:

T h e o r e m 9:

T h e o r e m I0:

Theo rem 11:

Theo rem 12:

Theo rem 13:

Theo rem 14:

Theorem 15:

The Socle as a Di rec t P r o d u c t

The Socle of a P r imi t ive Group is a Di rec t P r o d u c t of

I somorph ic S imple Groups

Poin t S tab i l i ze r S t r u c t u r e of the Nonabe l ian Socle of a

P r imi t ive Group

Degree Rela t ionsh ips for P r imi t ive Groups with Nona-

be t i an Socle

Po lynomia l Time

Polynomia l Time

Po lynomia l Time

Po lynomia l Time

Po lynomia l Time

Polynomia l Time

Po lynomia l Time

Equivalence of Problems i and 2

Equivalence of Problems 2 and 3

Reduc t ion of P r o b l e m 5 to P r o b l e m 2

Equivalence of P r o b l e m s 5 and 6




Po lynomia l Time Equivalence of P r o b l e m 8 with

P r o b l e m s 5 and 6

P r o b l e m 12 is in NPf~eoNP

I s o m o r p h i s m of Connec ted Label led Cycle Graphs

Testing I s o m o r p h i s m of Connec ted Label led

Graphs

Obtaining the Normal Closure

G / N is Abelian iff G' < N

Ni lpotence Cr i t e r ion

[A,B] as Normal Closure

Cycle

201

203

206

208

237

237

239

240

240

243

244

244

247

255

256

262

266

267

269

310

7. Corollary Index

Chapter II:

Corollary L: The Number of Graph Isomorphisms Equals the Order

of the Automorphism Group

Corollary 2: Automorphism Groups of Isomorphic Graphs are Conju-

gate

Corollary 8: Group Intersection is at least as Hard as Graph Isomor-

phism

T T, Chapter At,

Corol lary ,~.~

Corol lary 2:

Corol la ry 3:

Corol lary 4:

Corol lary 5:

Corollary 6:

Chapter IV:

Corollary 1:

Corollary 2:

Corol la ry 8:

Corol la ry 4:

Corol lary 5:

Corol la ry 6:

Corol lary 7:

Corollary 8:

Corol lary 9:

Generation of Uniformly Distributed Random Permuta-

tions in a Group G

Isomorphism of A(k)/A (k+1) with A, (k)

Algorithm 2 Determines a System of Imprimitivity

Determining a Minimal System of Imprimitivity and the

Induced Factor Group in a p-Group

Determining a Sylow p-Subgroup of S n Containing a

Given p-Group G < S n

induced p-Step Central Series in P1rbSn

If X is a Binary Cone Graph, t h e n Aub(X) is a 2-Group

Characterization of the Pointwise Stabilizer of

VoU " ' ° ~)Vk in Aute(Xk+1)

Timing of Automorphism of Binary Cone Graphs Rela-

tive to Setwise Stabilization in 2-Groups

Automorphism of Connected, Trivalent Graphs in 0(n zl)

Automorphism of Binary Cone Graphs in 0(n 14)

Automorphism of Connected, Trivalent Graphs in

O(nlS.!ogz(n))

Automorphism of Binary Cone Graphs in 0(nlZ.log2(n))

Composition Sequence for Set-wise Stabilizers of Maxi-

mal Sets of I m p r i m i t i v i t y

Conver t ing a Compos i t ion Sequence into an I m p r i m i -

t iv i ty Sequence

23

23

31

68

80

98

98

99

t04

!16

128

i;e8

L28

L38

!38

148

146

311

Chapter IV:

Corollary 10:

Corollary 11:

Corollary 12:

Corollary 13:

Corollary 14:

Corollary 15:

Corollary 16:

Chapter V:

Corollary l:

Corollary 2:

Corollary 3:

Corollary 4:

Corollary 5:

Corollary 6:

Corollary 7:

Corollary 8:

Chapter VI:

Corollary 1:

Corollary 2:

Corollary 3:

Corollary 4:

Corollary 5:

Corollary 6:

Determining a Point Stabilizer

Intersect ion of two p-Groups in O(n 3) Structure of the Group B

Automorphism of Extended Simple Cone Graphs

Intersection of a 2-Group with the Automorphism Group

of a Graph of Valence two

Timing of the Determination of Ae(X), X a Connected,

Trivalent Graph

Isomorphism of Trivalent Graphs in O(n 4)

Inn(G) is Isomorphic to G iff G has a Trivial Center

A Minimal Normal Subgroup is its own Socle

A Minimal Normal Subgroup is a Direct Product of Iso-

morphic Simple Groups

Structure of Aut(G), G a Direct Product of Isomorphic

Nonabelian Simple Groups

Order of Primitive Groups with Nonabelian Socle

Highest Power of q Dividing the Order of GL(m,p)

Index of Sylow p-Subgroups in Primitive Groups with

Abelian Socle

Index of Sylow p-Subgroups in Primitive Groups

Number of Nonisomorphic Graphs with n Vertices and p

Edges

Testing Isomorphism of Connected Labelled Cycle

Graphs in 0(ne.k)

Determining Aut(X) in O(n~.k), X a Connected Labelled

Cycle Graph

Determining Aut(X) in O(n~.k), X a Labelled Cycle Graph

Determining [A,B]

Determining whether G is Solvable or Nilpotent

147

157

159

170

170

175

176

196

201

201

205

209

213

214

215

234

255

255

256

269

269

INDICES

1. Problem Index

Chapter II:

Problem

Problem

Problem

Problem

Problem

Problem

Problem

Problem

Chapter IIi:

Problem

Problem

Problem

Problem

Problem

Problem

Chapter IV:

Problem

Problem

Problem

Chapter V:

Problem

Problem

Problem

Chapter Vh

Problem

Problem

Problem

i: Existence of Graph Isomorphism

2: Existence of Isomorphism of Labelled Graphs

3: Graph Isomorphism

4: Graph Automorphism

5: Order of the Automorphism Croup

6: Number of Isomorphisms

7: Group Specification by Generators

8: Determining an Accessible Group

i: Labelled Graph Automorphism

2: Automorphism of Cone Graphs of Fixed Degree

3: Sylow p-Subgroup of Sn Containing a Given p-Group

4: Minimal System of Imprimitivity for p-Groups

!nduced Factor Groups

5: p-Step Central Series for Sylow p-Subgroup of S n

5: Setwise Stabilizer in p-Groups

i: Trivalent Graph Aufiomorphism

2: Automorphism for Binary Cone Graphs

3: Imprimitivity Problem for 2-Groups

i: Automorphism of Graphs of Fixed Valence

8: Setwise Stabilizer in Groups in Fb

3: Sylow p-Subgroup of a Primitive Croup in F b

i: Double Coset Membership

2: Croup Facfiorization

3: Number of Factorizations

and

24

24

24

25

25

25

33

5i

6i

76

91

94

100

i08

1118

i16

i58

179

1188

215

236

28,6

236

297

Chapter VI:

Problem 4:

Problem 5:

Problem 6:

Problem 7:

Problem 8:

Problem 9:

Problem i0:

Problem i i :

Problem i2:

Pi 'oblem i3:

Problem i4:

Problem i5:

Problem i6:

Problem i7:

Problem i8:

Problem i9:

Problem 20:

Problem 2i:

Problem 22:

Problem 23:

Problem 24:

Problem 25:

Problem 26:

Problem 27:

Coset In tersec t ion Emptiness

Group In tersec t ion

Setwise Stabil izer

Centralizer in Another Croup

Rest r ic ted Graph Automorphism

Existence of Graph Isomorphism

Graph Automorphism

Small Graph with Prescr ibed Automorphism Croup

In tersec t ion o~ Commuting Groups

In tersec t ion with a Normalizing Croup

In tersec t ion with a p-Croup

Intersect ion with a Croup in F b

In tersec t ion of two Accessible Subgroups

In tersec t ion with an Accessible Subgroup

Centralizer in S n

Center

Intersection of the Automorphism Group of a Graph with

a Sylow p-Subgroup of Sym(V)

Normal Closure

Commutator Croup

Solvability

Nilpotenee

Double Coset Par t i t ion

Intransi t ive Subgroup Problem

Subspaee Problem

238

238

239

24i

24i

245

245

246

246

248

248

248

248

248

252

252

260

262

268

268

268

270

270

27i

298

2. Algori thm Index

Chapter II:

Algorithm ~:

Algorithm 2:

Algorithm 3:

Algorithm 4:

Algorithm 5:

Algorithm 6:

Algorithm 7:

Chapter III:

Algorithm i:

Algorithm 2:

Algorithm 3:

Chapter IV:

Membership Test in a Pe rmuta t ion Group G

Sift

Construct ion of a Representa t ion Matrix from Genera-

tors

Point Orbit and Schreier Vectors

Strong Generating Set for <K> Relative to [i ..... n]

Generalized Sift

Determination of a Subgroup Tower

Probabil is t ic Method for Problem I

Set of Imprimit iv i ty

Setwise Stabilizer in a p-Group (Method I)

Algorithm 1: Automorphism of a Connected, Trivalent Graph

Algorithm 2:

Algorithm 3:

Chapter V:

Algorithm i:

Algorithm 2:

Algorithm 3:

Chapter VI:

Algorithm 1:

Algorithm 2:

Algorithm 3:

Setwise Stabi l izer in a p-Group (Method 2)

In tersec t ion of two p-Groups

Automorphism of Graphs of Valence d

Subgroup with p-Group Action

Setwise Stabil izer in Fb

In te rsec t ion with a Normalizing Group

Central izer in S n

Normal Closure

34

38

39

46

48

53

54

7i

96

Iii

126

I34

153

182

220

250

259

264

3. Def in i t ion Index

299

Chapter II:

Definition 1:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Definition 9:

Definition 10:

Definition 11:

Definition 12:

Definition 13:

Definition 14:

Definition 15:

Definition 16:

Definition 17:

Definition 18:

Definition 19:

Definition 20:

Definition 21:

Definition 22:

Chapter tII:

Definition I:

Definition ~:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Group

Subgroup

Left Cosets, Right Cosets

Normal Subgroup

Group Homomorphism

Conjugates

Generating Set

Orbit in a Permutation Group, Orbit Length

Point Stabilizer

Pointwise Stabil izer

Setwise Stabi l izer

Simple, Undi rec ted Graph

Automorph i sm of a Graph

I somorphism ol two Graphs

Complemen t Graph

Polynomial Time Equivalence of Prob lems in NP

Vertex Labelled Graph

Represen ta t ion Matrix

Base of a P e r m u t a t i o n Group

Basic Orbit of a P e r m u t a t i o n Group

Strong Genera t ing Set

(k,c)-Accessibility of Groups

Cone Graph, Root of a Cone Graph

Regular Cone Graph

Degree of a Cone Graph

k-Automorphism of a Cone Graph

k-lsomorphism of Cone Graphs

Simple Extension

p-Group

Sylow p-Subgroup

12

14

14

15

16

t6

18

18

i9

ig

19

21

21

22

23

24

24

35

44

44

47

55

73

73

74

77

78

83

85

86

300

Chapter !II:

Definition 9:

Definition i0:

Definition i i:

Chapter IV:

Definition i:

Definition 2:

Definition 3:

Chapter V:

Definition I:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Definition 7:

Definition 8:

Definition 9:

Definition i0:

Definition i i:

Definition !2:

Definition i3:

Definition i4:

Definition 15:

Definition 16:

Definition 17:

Definition 18:

Chapter VI:

Definition i:

Definition 2:

Definition 3:

Definition 4:

Definition 5:

Definition 6:

Center of a Group

Central Series, p-Step Central Series

Wreath Product

Commutator

Composition Series, Imprimitivity series

Composition Sequence, Imprimitivity Sequence

Simple Group

Subnormal Series

Section

Maximal System of Imprimitivity

Gi-component of Subgroups of GIxG 2

Centralizer in Another Group

Norrnalizer in Another Group

Right and Left Regular Representation

Semiregular Group

Characteristic Subgroup

Characteristically Simple Group

Minimal Normal Subgroup

Socle

E l e m e n t a r y Abelian Group

Genera l Linear Group

Full Diagona! Subgroup

Order of a Modulo p

Forward Connect iv i ty in Graphs

Double Coset

Cycle Graph of a Permutation

Labelled Cycle Graph of a Set of Permutations

Cycle Distance

Normal Closure

Commutator

86

87

88

140

143

144

184

i84

186

!89

i94

i95

195

197

i97

199

2OO

20O

2OO

203

2O4

2O7

211

228

232

253

254

254

262

265

301

Chapter VI:

Definition

Definition

Definition

7: Commuta tor Group, Derived Group

8: Solvable Group

9: Nilpotent Group

265

266

267

4. L e m m a Index

302

Chapter ti:

L e m m a

L e m m a

L e m m a

Lemma

L e m m a

L e m m a

L e m m a

Lemma

L e m m a 9:

Chapter IIh

L e m m a 1:

L e m m a 2:

L e m m a 3:

L e m m a 4:

L e m m a 5:

L e m m a 6:

Lemma 7:

L e m m a 8:

L e m m a 9:

L e m m a 10:

L e m m a i i :

L e m m a 12:

L e m m a 13:

L e m m a t4:

L e m m a 15:

L e m m a 16:

!: Polynomial Time Reduction of Problem 2 to Problem 1


3: Complete Right Transversals of a Point Stabilizer Sub-

group Tower Define Generating Sets


5: Polynomial. Time Reduction of Problem 5 to Problem 4

6: Polynomial Time Reduction of Problem i to Problem 5

7: Rows in a Representation Matrix are Right Transversals

8: Comple te Right Transversa ls in a Subgroup Tower Define

Genera t ing Sets

Inequal i ty of Indices

A Group from which Aut(X) is Accessible, X a Graph as in

Problem I

Bound on the Subgroup Indices above Aut(X)

Bound on the Subgroup Indices below Aut(X)

Generating Uniformly Distributed Random Integers

Random Sift

Probability of Obtaining a Complete Right Transversal by

Random Sift

Normality of A (k+1) in A (k)

A Group Containing Aut(X), X a Regular Cone Graph

Membership in A, (k)

Finiteness of Algorithm 2

Checking a System of Imprimitivity from a Generating

Set

p-Step Central Series for Cp

p-Step Central Series for a Direct Product

Subgroup Structure of A%B

A Central Subgroup of PI%Cp

CommutLng Properties of h i

25

26

27

28

29

29

36

,5!

56

64

64

65

67

69

69

78

82

84

96

97

i00

I00

101

10E

!03

303

Chapter III:

Lemma 17:

Lemma i8:

Lemma i9:

Lemma 20:

Lemma 21:

Chapter IV:

Lemma i:

Lemma 2:

Lemma 3:

Lemma 4:

Lemma 5:

Lemma 6:

Lemma 7:

Lemma 8:

Lemma 9:

Lemma 10:

Lemma 1 i:

Lemma i2:

Lemma i3:

Chapter V:

Lemma

ae m m a

Lemma

Lemma

Lemma

Lemma

Lemma

Lemma

Lemma

Properties of the Croups G (i'D

Properties of the Croups G (i'j)

If A normalizes B, then AB is a group

The order of AB

Setwise Stabilizers in an External Direct Product

Automorphisms Respect Family Types

Conditions for Extending a c Aute(Xk)

Generating Set for Aute(Xk+l)

Coset Intersection

Maximal Systems of Imprimitivity in p-Groups

Timing of Algorithm 2 for Transitive p-Croups

Schreier Generating Set for Subgroups

Elementary Properties of J(Afi,A)

Recursion Rules for J(A~,A)

PasUn~ Cosets J(A'~i~A )

On Point Stabilizers in the 3ocle of Primitive Croups

Imprimitivity Structure of Pair Action

Deletion of Redundant Generators from a Composition

Sequence

i: Composition Series Induced by Normal Silbgroup and

Factor Group

2: Characterization of F b

3: Closure Properties of F b

4: Composition Series of Symmetric Groups

5: Internal Direct Product

6: Inn(GlxCs) ~ Inn(G1)xInn(C~)

7: A Regular Croup is the Right (Left) Regular Representa-

tion of Another Group

8: On Centralizers of Minimal Normal Subgroups of Primi-

tive Groups

9: Normalizer of a Full Diagonal Subgroup

103

i04

106

107

108

i2i

122

123

i30

i3i

137

i42

148

i48

t52

i59

160

i69

i85

i86

i87

i87

i93

i96

i97

202

207

304

Chapter V:

Lemma i0:

Lemma i l :

Lemma 12:

Lemma I3:

Lemma i4:

Lemma i5:

Chap t e r VI:

Lemma l:

Lemma 2:

Lemma 3:

Lemma 4:

Lemma 5:

Lemma 6:

Lemma 7:

Lemma 8:

Lemma 9:

Lemma i0:

Lemma i i :

Lemma i2:

Lemma ].3:

Subgroup S t r u c t u r e of Direct P roduc t s of Isomorphic

Nonabel ian Simple Groups

On Point Stabilizers of the Socle

(i+pSy) m = l+pSu

Highest Power of p Dividing (pt)

(i+pSy) pt = i+pS+tu with Except ions

Number of Subprob tems when Dete rmin ing Setwise Sta-

bil izers in p-Groups

Cardinai i ty of A~B

Number of Fac tor iza t ions in AB

Polynomial Time Equivalence of Prob lems 4 and 2

The Centra l izer as I n t e r s ec t i on of Genera to r Centra l izers

Centra l izer of a P e r m u t a t i o n

Necessary and Sufficient Conditions that AB is a Group

Sufficient Condition that AB is a Group

Point Stabilizers Inherit the Normalizing Property

Automorphism Group of a Cycle Graph

Automorphism Group of a Labelled Cycle Graph

Properties of Commutators

Properties of Commutator Groups

Commutator Properties of Central Series

207

208

211

211

2 i2

224

332

237

239

242

242

247

247

249

253

254

265

265

267

5. Proposition Index

Chapter Ih

Proposition

Proposition

Proposition

Proposition

Proposition

Chapter III:

Proposition

Chapter IV:

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

Chapter V:

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

Proposition

305


2: Correctness and Timing of Algorithms 2 and 3

3: Correctness and Timing of Algorithm 4



1: St ructure of the Wreath Product of Symmetr ic

Croups

1: Determining a Nontriviai Sys tem of Imprimitivity in a

p-Group

2: Determining a Complete Imprimitivity St ructure for

a p-Group

3: Determining an Imprimitivity Sequence f rom

Generators

4: Setwise Stabilizer in a p-Group in O(n 3)

5: Determining a Sylow 2-SubgroupPof Sym(L 2) Contain-

ing the Pair Action of the 3-Group G < Sym(L)

6: Determining Auto(X) in O(nS), X a Simple Cone Graph

l: Cardinality of Elements with a Fixed Gl-component

2: Inn(G) is Normal in Aut(G)

3: Inn(G) is Faithful on a Normal Subgroup N iff

CdN) = I

4: The Centralizer of a Regular Group G is Isomorphic to

G

5: St ructure of *GxG* and Point Stabilizers

6: Minimal Normal Subgroups are Characterist ically

Simple

7: The Socle is a Characterist ic Subgroup

34

40

45

55

55

89

141

143

144

157

163

167

195

196

196

198

198

200

201

306

Chapter V:

Proposition 8:

Proposition 9:

Proposition 10:

Proposition 11:

Proposition i2:

Proposition !3:

Proposition i4:

Proposition i5:

Chapter VI:

Proposition i:

Proposition 2:

Proposition 3:

Proposition 4:

Proposition 5:

Proposition 6:

Proposition 7:

Proposition 8:

Proposition 9:

Nontrivial Normal Subgroups of Primitive Groups are

Transitive

Order of GL(m,p)

Normal Structure of Direct Products of Isomorphic

Nonabe!ian Simple Croups

A Primitive Group Can Be Embedded into the Auto-

morphism Croup of its Nonabelian Soele

The Abelian Soele of a Primitive Group is Elementary

Abelian

identities for (i +pSy) ptm

Timing of Algorithm 2 for Groups in F b

Timing of Algorithm 3 for Groups in Fb

Graph Isomorphism is a Double Coset Problem


Testing whether B Normalizes A

Intersection with a p-Group

Timing of Algorithm

Determining the Center in O(IKI "nZ+n a)

Problem 20 is in P


Solvability Criterion

202

204

204

206

2i0

212

217

225

233

~5I

251

251

259

26O

261

264

266

307

6. T h e o r e m Index

C h a p t e r I I :

Theo rem 1:

Theo rem 2:

Theo rem 3:

T h e o r e m 4:

T h e o r e m 5:

Theo rem 6:

Theorem 7:

T h e o r e m 8:

Theo rem 9:

Theorem lO:

Theo rem l l :

Theo rem i2:

T h e o r e m iS:

Theo rem i4:

C h a p t e r III:

T h e o r e m 1:

T h e o r e m 2:

Theo rem 8:

Theo rem 4:

Theo rem 5:

T h e o r e m 6:

T h e o r e m 7:

T h e o r e m 8:

Theo rem 9:

Lag range ' s T h e o r e m

Cayley 's Theo rem

C o r r e s p o n d e n c e of Orbi t Poin ts wi th Right Cosets of

Po in t S tab i l i ze r

The I s o m o r p h i s m s of a Graph F o r m a Right Coset of t he

A u t o m o r p h i s m Group

A Bound on the Minimal N u m b e r of G e n e r a t o r s

Po lynomia l Time Equivalence of P r o b l e m s i t h r o u g h 6

The A u t o m o r p h i s m Group of a Graph is the I n t e r s e c t i o n

of two Known P e r m u t a t i o n Groups

Necessary and Sufficient Conditions for a Representa-

tion Matrix to Specify a Group

P r o b l e m 7 is in P

Exis tence of a Smal l S t rong Genera t ing Se t

N e c e s s a r y and Sufficient Condi t ions for the Right

Transversa l s of a Subgroup Tower to be Comple te

P r o b l e m 8 is in P for (k ,c ) -Access ib le Groups

I n t e r s e c t i o n of two (k ,e) -Access ib le Groups

I n t e r s e c t i o n of a (k ,c) -Access ib le Group with a P e r m u t a -

t ion Group with Known C e n e r a t o r s

De t e rmin i s t i c Solu t ion of P r o b l e m i

Gene ra t i on of Uniformly D i s t r i bu t ed Random P e r m u t a -

t ions in S n

Random Algorithm for Determining a Subgroup Tower

Timing of Algor i thm 1

C h a r a c t e r i z a t i o n of A(k)/A (k+])

Cauchy ' s T h e o r e m

Sylow's Theo rems

Every p-Group has a Nontr ivial Cen te r

Every p-Group has a p -S tep Cent ra l Ser ies

14

i ?

19

22

25

29

3i

35

40

49

52

56

56

57

65

67

70

7 i

80

86

86

87

87

308

Chapter I I I :

Theorem lO:

Theorem l l :

Theorem t~:

Theorem l3:

Theorem l4:

Theorem l5:

Theorem l6:

Theorem I7:

Theorem l8:

Chapter IV:

Theorem i:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 6:

Theorem 7:

Theorem 8:

Theorem 9:

Chap te r V:

Theorem i:

Theorem 2:

Theorem 3:

Theorem 4:

Theorem 5:

Theorem 8:

Struct-~e of Sylow p-Subgroups of S n

Characterization of Primitivity and Imprimitivity

The Imprimitivity Sets of p-Groups

Loop Invariant for Algorithm 2

Correctness of Algorithm


Determining a p-Step Central Series of Sylow p-

Subgroups of Sn

Determining Setwise Stabilizers in Sylow p-Subgroups of

Sn

Correctness and Timing of Algorithm 3

If X is Connected and Trivalent, then Aura(X) is a 2-

Group

Timing of Automorphism of Connected, Trivalent Graphs

Relative to Setwise Stabihzation in 2-Groups

Determining Setwise Stabilizers in p-Groups

Lifting the Stabilizer of a Maximal Set of Imprimitivity

Determining J((GxH)K,X) in O(n 3)

Solution of the Imprimitivity Problem for 2-Groups in

O(nD Gadget Reduction for Extended Simple Cone Graphs

Determining the Elements of Ae(Xk) which may be

Extended to an Automorphism of Xk+ 1

Timing of the Iterative Step in Comput ing Ae(X )

Timing of Problem i Relative to Setwise Stabilization

Jordan-HiJlder' s Theorem

Composition Series of Subgroups

AULe(X ) E Ff(d), where X is Connected of Valence d

The Centralizer of a Transitive Group is Semiregular

Conjugation of a Regular Normal Subgroup Under Ele-

ments of a Point Stabilizer

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92

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98

98

i06

l lO

i12

l t 5

128

138

145

157

i65

169

!74

i75

184

i85

185

187

t98

!99

309

C h a p t e r V:

T h e o r e m 7:

T h e o r e m 8:

Theo rem 9:

Theo rem 10:

C h a p t e r VI:

T h e o r e m 1:

Theo rem 2:

T h e o r e m 3:

Theo rem 4:

Theo rem 5:

Theo rem 6:

T h e o r e m 7:

Theo rem 8:

T h e o r e m 9:

T h e o r e m I0:

Theo rem 11:

Theo rem 12:

Theo rem 13:

Theo rem 14:

Theorem 15:

The Socle as a Di rec t P r o d u c t

The Socle of a P r imi t ive Group is a Di rec t P r o d u c t of

I somorph ic S imple Groups

Poin t S tab i l i ze r S t r u c t u r e of the Nonabe l ian Socle of a

P r imi t ive Group

Degree Rela t ionsh ips for P r imi t ive Groups with Nona-

be t i an Socle

Po lynomia l Time

Polynomia l Time

Po lynomia l Time

Po lynomia l Time

Po lynomia l Time

Polynomia l Time

Po lynomia l Time

Equivalence of Problems i and 2

Equivalence of Problems 2 and 3


Equivalence of P r o b l e m s 5 and 6




Po lynomia l Time Equivalence of P r o b l e m 8 with

P r o b l e m s 5 and 6

P r o b l e m 12 is in NPf~eoNP

I s o m o r p h i s m of Connec ted Label led Cycle Graphs

Testing I s o m o r p h i s m of Connec ted Label led

Graphs

Obtaining the Normal Closure

G / N is Abelian iff G' < N

Ni lpotence Cr i t e r ion

[A,B] as Normal Closure

Cycle

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269

310

7. Corollary Index

Chapter II:

Corollary L: The Number of Graph Isomorphisms Equals the Order

of the Automorphism Group

Corollary 2: Automorphism Groups of Isomorphic Graphs are Conju-

gate

Corollary 8: Group Intersection is at least as Hard as Graph Isomor-

phism

T T, Chapter At,

Corol lary ,~.~

Corol lary 2:

Corol la ry 3:

Corol lary 4:

Corol lary 5:

Corollary 6:

Chapter IV:

Corollary 1:

Corollary 2:

Corol la ry 8:

Corol la ry 4:

Corol lary 5:

Corol la ry 6:

Corol lary 7:

Corollary 8:

Corol lary 9:

Generation of Uniformly Distributed Random Permuta-

tions in a Group G

Isomorphism of A(k)/A (k+1) with A, (k)

Algorithm 2 Determines a System of Imprimitivity

Determining a Minimal System of Imprimitivity and the

Induced Factor Group in a p-Group

Determining a Sylow p-Subgroup of S n Containing a

Given p-Group G < S n

induced p-Step Central Series in P1rbSn

If X is a Binary Cone Graph, t h e n Aub(X) is a 2-Group

Characterization of the Pointwise Stabilizer of

VoU " ' ° ~)Vk in Aute(Xk+1)

Timing of Automorphism of Binary Cone Graphs Rela-

tive to Setwise Stabilization in 2-Groups

Automorphism of Connected, Trivalent Graphs in 0(n zl)

Automorphism of Binary Cone Graphs in 0(n 14)

Automorphism of Connected, Trivalent Graphs in

O(nlS.!ogz(n))

Automorphism of Binary Cone Graphs in 0(nlZ.log2(n))

Composition Sequence for Set-wise Stabilizers of Maxi-

mal Sets of I m p r i m i t i v i t y

Conver t ing a Compos i t ion Sequence into an I m p r i m i -

t iv i ty Sequence

23

23

31

68

80

98

98

99

t04

!16

128

i;e8

L28

L38

!38

148

146

311

Chapter IV:

Corollary 10:

Corollary 11:

Corollary 12:

Corollary 13:

Corollary 14:

Corollary 15:

Corollary 16:

Chapter V:

Corollary l:

Corollary 2:

Corollary 3:

Corollary 4:

Corollary 5:

Corollary 6:

Corollary 7:

Corollary 8:

Chapter VI:

Corollary 1:

Corollary 2:

Corollary 3:

Corollary 4:

Corollary 5:

Corollary 6:

Determining a Point Stabilizer

Intersect ion of two p-Groups in O(n 3) Structure of the Group B

Automorphism of Extended Simple Cone Graphs

Intersection of a 2-Group with the Automorphism Group

of a Graph of Valence two

Timing of the Determination of Ae(X), X a Connected,

Trivalent Graph

Isomorphism of Trivalent Graphs in O(n 4)

Inn(G) is Isomorphic to G iff G has a Trivial Center

A Minimal Normal Subgroup is its own Socle

A Minimal Normal Subgroup is a Direct Product of Iso-

morphic Simple Groups

Structure of Aut(G), G a Direct Product of Isomorphic

Nonabelian Simple Groups

Order of Primitive Groups with Nonabelian Socle

Highest Power of q Dividing the Order of GL(m,p)

Index of Sylow p-Subgroups in Primitive Groups with

Abelian Socle

Index of Sylow p-Subgroups in Primitive Groups

Number of Nonisomorphic Graphs with n Vertices and p

Edges

Testing Isomorphism of Connected Labelled Cycle

Graphs in 0(ne.k)

Determining Aut(X) in O(n~.k), X a Connected Labelled

Cycle Graph

Determining Aut(X) in O(n~.k), X a Labelled Cycle Graph

Determining [A,B]

Determining whether G is Solvable or Nilpotent

147

157

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170

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201

201

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215

234

255

255

256

269

269

1978] wesley (1978) - purdue universityk. k. agarwal [1978] "transforma2icn and canonization...

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