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
The Structure of a Primitive Group | 106 |
Bounds on Orders of Permutation Groups | 143 |
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 |
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Common terms and phrases
2-cycles action of G acts transitively Algebra Alt(N Aut(T automorphism group conjugacy class conjugation construction cosets countable cycle define denote diagonal elementary abelian elements example Exercise exists Fano subplane finite group finite primitive group finite simple groups free group Frobenius group G acts G is 2-transitive G is primitive graph group acting group G group of degree Hence highly transitive homomorphic hyperovals hypothesis imprimitive induces infinite integer isomorphic Jordan complement Jordan group Lemma Let G mapping Mathieu groups maximal subgroup minimal degree minimal normal subgroup nontrivial p-group pair permutation isomorphic point stabilizer prime primitive permutation groups PROOF prove quadrangle regular Sect shows that G soc(G solvable Steiner system subgroup of G subnormal subgroup subset subspaces Suppose that G Sylow Sylow p-subgroup symmetric group T₁ Theorem transitive extension transitive group unique vertex vertices wreath product