Permutation GroupsPermutation Groups form one of the oldest parts of group theory. Through the ubiquity of group actions and the concrete representations which they afford, both finite and infinite permutation groups arise in many parts of mathematics and continue to be a lively topic of research in their own right. The book begins with the basic ideas, standard constructions and important examples in the theory of permutation groups.It then develops the combinatorial and group theoretic structure of primitive groups leading to the proof of the pivotal O'Nan-Scott Theorem which links finite primitive groups with finite simple groups. Special topics covered include the Mathieu groups, multiply transitive groups, and recent work on the subgroups of the infinite symmetric groups. This text can serve as an introduction to permutation groups in a course at the graduate or advanced undergraduate level, or for self- study. It includes many exercises and detailed references to the current literature. |
Contents
Examples and Constructions | 33 |
The Action of a Permutation Group | 65 |
The Structure of a Primitive Group | 106 |
Bounds on Orders of Permutation Groups | 143 |
Group | 159 |
The Mathieu Groups and Steiner Systems | 177 |
Multiply Transitive Groups | 210 |
The Structure of the Symmetric Groups | 255 |
Examples and Applications of Infinite Permutation | 274 |
Appendix A Classification of Finite Simple Groups | 302 |
| 327 | |
| 341 | |
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Common terms and phrases
2-cycles 2-transitive action of G acts transitively Algebraic automorphism group block for G conjugacy class conjugation construction cosets countable cycle define denote diagonal elementary abelian elements example Exercise exists finite group finite primitive group finite simple groups fixed points free group Frobenius group G acts G contains G is 2-transitive G is primitive geometry graph group acting group G group of degree Hence homomorphism hyperovals hypothesis imprimitive induces infinite integer isomorphic Jordan complement Lemma Let G mapping Mathieu groups minimal degree minimal normal subgroup nontrivial p-group pair permutation groups permutation isomorphic point stabilizer polynomial prime PROOF prove quadrangle Sect show that G soc(G socle Steiner system subgroup of G subnormal subgroup subset subspace Suppose that G Sylow p-subgroup symmetric group T₁ Theorem transitive extension transitive group transitive subgroup unique vertices wreath product Ε Ω