Permutation Groups

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Cambridge University Press, Feb 4, 1999 - Mathematics - 220 pages
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Permutation groups are one of the oldest topics in algebra. Their study has recently been revolutionized by new developments, particularly the Classification of Finite Simple Groups, but also relations with logic and combinatorics, and importantly, computer algebra systems have been introduced that can deal with large permutation groups. This text summarizes these developments, including an introduction to relevant computer algebra systems, sketch proofs of major theorems, and many examples of applying the Classification of Finite Simple Groups. It is aimed at beginning graduate students and experts in other areas, and grew from a short course at the EIDMA institute in Eindhoven.
 

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Contents

General theory
1
12 Actions and Gspaces
2
13 Orbits and transitivity
3
14 Transitive groups and coset spaces
4
15 Sylows Theorem
6
16 Regular groups
7
17 Groups with regular normal subgroups
9
18 Multiple transitivity
10
44 Some basic groups
104
45 The ONanScott Theorem
105
46 Maximal subgroups of Sₙ
107
47 The finite simple groups
108
Multiplytransitive groups
110
Degrees of primitive groups
111
Orders of primitive groups
112
The length of Sₙ
117

19 Primitivity
11
111 Orbitals
13
112 Sharp ktransitivity
15
113 The SchreierSims algorithm
17
114 Jerrums filter
19
115 The length of Sₙ
20
116 At the keyboard
21
Cycles and parity
25
118 Exercises
27
Representation theory
35
22 Centraliser algebra
36
23 The OrbitCounting Lemma
37
24 Character theory
40
25 The permutation character
42
26 The diagonal group
44
27 FrobeniusSchur index
45
28 Parkers Lemma
47
29 Characters of abelian groups
51
210 Characters of the symmetric group
53
Möbius inversion
56
212 Exercises
57
Coherent configurations
63
32 Algebraic theory
66
33 Association schemes
68
34 Algebra of association schemes
73
Strongly regular graphs
76
36 The HoffmanSingleton graph
79
37 Automorphisms
83
38 Valency bounds
85
39 Distancetransitive graphs
87
310 Multiplicity bounds
90
311 Duality
92
312 Wielandts Theorem
94
313 Exercises
95
The ONanScott Theorem
99
42 Precursors
100
43 Product action and basic groups
102
Distancetransitive graphs
118
413 Bases
120
414 Geometric groups and IBIS groups
123
Matroids
125
416 Exercises
127
Oligomorphic groups
131
52 Oligomorphic groups
135
54 Automorphism groups and topology
138
55 Countably categorical structures
139
56 Homogeneous structures
141
57 Cycle index
143
58 A graded algebra
145
59 Monotonicity
147
510 Settransitive groups
148
511 Growth rates
150
512 On complementation and switching
153
Cycle index
157
514 Exercises
160
Miscellanea
165
62 Neumanns Lemma
167
63 Cofinitary permutation groups
168
64 Theorems of Blichfeldt and Maillet
170
65 Cycleclosed permutation groups
173
67 The OrbitCounting Lemma revisited
176
68 Jordan groups
179
69 Orbits on moieties
182
610 Exercises
183
Tables
187
71 Simple groups of Lie type
188
72 Sporadic simple groups
192
73 Affine 2transitive groups
194
74 Almost simple 2transitive groups
196
75 Exercises
198
Bibliography
199
Index
213
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