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E-raamat: Probabilistic Group Theory, Combinatorics, and Computing: Lectures from the Fifth de Brun Workshop

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Probabilistic Group Theory, Combinatorics and Computing is based on lecture courses held at the Fifth de Brún Workshop in Galway, Ireland in April 2011. Each course discusses computational and algorithmic aspects that have recently emerged at the interface of group theory and combinatorics, with a strong focus on probabilistic methods and results. The courses served as a forum for devising new strategic approaches and for discussing the main open problems to be solved in the further development of each area. The book represents a valuable resource for advanced lecture courses. Researchers at all levels are introduced to the main methods and the state-of-the-art, leading up to the very latest developments. One primary aim of the books approach and design is to enable postgraduate students to make immediate use of the material presented.
1 Probabilistic and Asymptotic Aspects of Finite Simple Groups
1(34)
Martin W. Liebeck
1.1 Random Generation of Simple Groups and Maximal Subgroups
1(10)
1.1.1 Alternating Groups
1(3)
1.1.2 Groups of Lie Type
4(4)
1.1.3 Other Results on Random Generation
8(2)
1.1.4 Generation of Maximal Subgroups
10(1)
1.2 Random Generation of Arbitrary Finite Groups
11(4)
1.3 Representation Varieties and Character-Theoretic Methods
15(11)
1.3.1 Fuchsian Groups
16(1)
1.3.2 Character Theory
17(2)
1.3.3 Symmetric and Alternating Groups
19(1)
1.3.4 Groups of Lie Type
20(1)
1.3.5 Representation Varieties
21(2)
1.3.6 Triangle Groups
23(3)
1.4 Cayley Graphs of Simple Groups: Diameter and Growth
26(9)
1.4.1 Conjugacy Classes
28(1)
1.4.2 Babai's Conjecture
29(2)
References
31(4)
2 Estimation Problems and Randomised Group Algorithms
35(48)
Alice C. Niemeyer
Cheryl E. Praeger
Akos Seress
2.1 Estimation and Randomization
35(7)
2.1.1 Computation with Permutation Groups
35(1)
2.1.2 Recognising the Permutation Group Giants
36(1)
2.1.3 Monte Carlo Algorithms
37(2)
2.1.4 What Kinds of Estimates and in What Groups?
39(1)
2.1.5 What Group is That: Recognising Classical Groups as Matrix Groups
39(3)
2.1.6 What Group is That: Recognising Lie Type Groups in Arbitrary Representations
42(1)
2.2 Proportions of Elements in Symmetric Groups
42(15)
2.2.1 Notation
42(1)
2.2.2 Historical Notes
42(1)
2.2.3 Orders of Permutations
43(1)
2.2.4 Number of Cycles
44(1)
2.2.5 Generating Functions
44(4)
2.2.6 Solutions to xm = 1 in Symmetric and Alternating Groups
48(3)
2.2.7 The Munchausen Method (Bootstrapping)
51(3)
2.2.8 Algorithmic Applications of Proportions in Symmetric Groups
54(2)
2.2.9 Restrictions on Cycle Lengths
56(1)
2.3 Estimation Techniques in Lie Type Groups
57(11)
2.3.1 p-Singular Elements in Permutation Groups
57(1)
2.3.2 Quokka Subsets of Finite Groups
58(1)
2.3.3 Estimation Theory for Quokka Sets
59(2)
2.3.4 Strong Involutions in Classical Groups
61(2)
2.3.5 More Comments on Strong Involutions
63(2)
2.3.6 Regular Semisimple Elements and Generating Functions
65(3)
2.4 Computing Centralisers of Involutions
68(15)
2.4.1 Applications of Centralisers of Involutions Computations
69(1)
2.4.2 Constructive Membership in Lie Type Groups
70(2)
2.4.3 Constructive Recognition of Lie Type Groups
72(2)
2.4.4 Computation of an Element Centralising an Involution
74(1)
2.4.5 Computation of the Full Centraliser
75(3)
References
78(5)
3 Designs, Groups and Computing
83
Leonard H. Soicher
3.1 Introduction
83(1)
3.2 Background Material
84(5)
3.2.1 Block Designs
84(1)
3.2.2 Efficiency Measures of 1-Designs
85(1)
3.2.3 Permutation Groups
86(1)
3.2.4 Latin Squares
87(1)
3.2.5 Semi-Latin Squares
88(1)
3.3 Optimality Results for Semi-Latin Squares
89(1)
3.4 Uniform Semi-Latin Squares
90(1)
3.5 Semi-Latin Squares from Transitive Permutation Groups
91(3)
3.5.1 The Canonical Efficiency Factors of SLS(G)
92(2)
3.6 The Design Package
94(4)
3.6.1 The Block Designs Function
95(2)
3.6.2 The Block Design Efficiency Function
97(1)
3.7 Classifying Semi-Latin Squares
98(4)
3.8 Efficient Semi-Latin Squares as Subsquares of Uniform Semi-Latin Squares
102(3)
3.9 Some Open Problems
105
References
106
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