Ebook: An Introduction to the Theory of Groups

Groups and Homomorphisms -- The Isomorphism Theorems -- Symmetric Groups and G-Sets -- The Sylow Theorems -- Normal Series -- Finite Direct Products -- Extensions and Cohomology -- Some Simple Linear Groups -- Permutations and Mathieu Groups -- Abelian Groups -- Free Groups and Free Products -- The Word Problem -- Appendices: Some Major Algebraic Systems -- Equivalence Relations and Equivalence Classes -- Functions -- Zorn's Lemma -- Countability -- Commutative Rings -- Bibliography -- Notation -- Index.;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).