This book is translated from the second Russian edition and with added notes by K.A. Hirsch. ""Teoriya Grupp"" by Kurosh was widely acclaimed, in its first edition, as the first modern text on the general theory of groups, with the major emphasis on infinite groups. The decade that followed brought about a remarkable growth and maturity in the theory of groups, so that this second edition, in English translation, represents a complete rewriting of the first edition.The book can be used as a beginning text, the only requirement being some mathematical maturity and a knowledge of the elements of transfinite numbers. Many new sections were added to this second edition, and many old ones were completely revised: the theory of abelian groups was significantly revised; many significant additions were made to the section on the theory of free groups and free products; an entire chapter is devoted to group extensions; and the deep changes in the theory of solvable and nilpotent groups - one of the large and rich branches of the theory of groups - are covered in this work. Each volume concludes with Editor's Notes and a Bibliography.
Part Three. Group-Theoretical Constructions: Free Products and Free Groups:; 9.33 Definition of a free product; 9.34 Subgroups of a free product; 9.35 Isomorphism of free decompositions. Free products with an amalgamated subgroup; 9.36 Subgroups of free groups; 9.37 Fully invariant subgroups of free groups. Identical relations Finitely Generated Groups:; 10.38 General properties of finitely generated groups; 10.39 Grusko's theorem; 10.40 Grusko's theorem (conclusion); 10.41 Groups with a finite number of defining relations Direct Products. Lattices:; 11.42 Preliminary remarks; 11.43 Lattices; 11.44 Modular and complete modular lattices; 11.45 Direct sums in complete modular lattices; 11.46 Further lemmas; 11.47 The fundamental theorem Extensions of Groups:; 12.48 Factor systems; 12.49 Extensions of abelian groups. Cohomology groups; 12.50 Calculation of the second cohomology group; 12.51 Extensions of non-commutative groups; 12.52 Special cases; Part Four. Solvable and Nilpotent Groups: Finiteness Conditions, Sylow Subgroups, and Related Problems:; 13.53 Finiteness conditions; 13.54 Sylow subgroups. The centers of $p$-groups; 13.55 Local properties; 13.56 Normal and invariant systems Solvable Groups:; 14.57 Solvable and generalized solvable groups; 14.58 Local theorems. Locally solvable groups; 14.59 Solvable groups with finiteness conditions; 14.60 Sylow $\Pi$-subgroups of solvable groups; 14.61 Finite semi-simple groups Nilpotent Groups:; 15.62 Nilpotent and finite nilpotent groups; 15.63 Generalized nilpotent groups; 15.64 Connections with solvable groups. $S$-groups. Finiteness conditions; 15.65 Complete nilpotent groups; 15.66 Groups with unique extraction of roots; 15.67 Locally nilpotent torsion-free groups Appendixes Bibliography Author Index Subject Index.