An introduction to the theory of groups
Anyone who has studied "abstract algebra" and linear algebra as an undergraduate can understand this book. This edition has been completely revised and reorganized, without however losing any of the clarity of presentation that was the hallmark of the previous editions.The first six chapters provide ample material for a first course: beginning with the basic properties of groups and homomorphisms, topics covered include Lagrange's theorem, the Noether isomorphism theorems, symmetric groups, "G"-sets, the Sylow theorems, finite Abelian groups, the Krull-Schmidt theorem, solvable and nilpotent groups, and the Jordan-Holder theorem.The middle portion of the book uses the Jordan-Holder theorem to organize the discussion of extensions (automorphism groups, semidirect products, the Schur-Zassenhaus lemma, Schur multipliers) and simple groups (simplicity of projective unimodular groups and, after a return to "G"-sets, a construction of the sporadic Mathieu groups).
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THE ISOMORPHISM THEOREMS
THE SYLOW THEOREMS
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affine Assume Aut(K automorphism commute composition series conjugacy classes conjugate Corollary cosets cyclic groups defined Definition Let denoted diagram direct product direct sum disjoint divisor equation equivalence exact sequence example Exercise exists factor groups factor set field finite group finitely presented group fixes follows free abelian free abelian group free group function functor G acts G contains G-set group G group of order hence Hint HNN extension homomorphism imbedded implies induction infinite integer kernel Lemma Let G Let H linear matrix multiplicative group nilpotent nonsingular normal series normal subgroup notation number of elements one-one correspondence p-group p-primary permutation polynomial positive words prime Proof Let quotient reader recursive relations semidirect product semigroup shows simple group solvable group stable letter Steiner system subgroup H subgroup of G subgroup of order subset subword summand Sylow p-subgroup torsion-free transitive G-set transvection unique vector space