The first six chapters of this book ample material for a first course: beginning with the basic properties of groups and homomorphisms, the topics covered include Lagrange’s theorem, the Noether isomorphism theorems, symmetric groups, G-sets, the Sylow theorems, finite abelian groups, Krull-Schmidt theorem, solvable and nilpotent groups, and the Jordan-Holder theorem. The middle portion of the book then 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). The book closes with three chapters on: infinite abelian groups, with emphasis on countable groups; free groups and presentations of groups, coset enumeration, free products, amalgams, and HNN extensions; and a complete proof of the unsolvability of the word problem for finitely presented groups, as well as the Higman imbedding theorem and the undecidability of the isomorphism problem for groups.