Permutation 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.
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2-cycles action of G acts transitively affine group Algebraic automorphism group block for G conjugacy class conjugation construction cosets countable cycle define denote diagonal disjoint elementary abelian exactly example exists Fano subplane fc-transitive finite group finite primitive group finite simple groups fixed points free group Frobenius group G contains G is 2-transitive G is primitive geometry graph Graph(A group acting 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 orbits of G p-group pair permutation groups permutation isomorphic point stabilizer polynomial prime Proof prove quadrangle representation Sect show that G soc(G socle solvable Steiner system subgroup of G subnormal subgroup subset subspace Suppose that G Sylow p-subgroup Sym(Cl Sym(Q symmetric group Theorem transitive extension transitive group transitive subgroup unique vertices Wielandt wreath product