The present edition differs from the first in several places. In particular our treatment of polycyclic and locally polycyclic groups-the most natural generalizations of the classical concept of a finite soluble group-has been expanded. We thank Ju. M. Gorcakov, V. A. Curkin and V. P. Sunkov for many useful remarks. The Authors Novosibirsk, Akademgorodok, January 14, 1976. v Preface to the First Edition This book consists of notes from lectures given by the authors at Novosi­ birsk University from 1968 to 1970. Our intention was to set forth just the fundamentals of group theory, avoiding excessive detail and skirting the quagmire of generalizations (however a few generalizations are nonetheless considered-see the last sections of Chapters 6 and 7). We hope that the student desiring to work in the theory of groups, having become acquainted with its fundamentals from these notes, will quickly be able to proceed to the specialist literature on his chosen topic. We have striven not to cross the boundary between abstract and scholastic group theory, elucidating difficult concepts by means of simple examples wherever possible. Four types of examples accompany the theory: numbers under addition, numbers under multiplication, permutations, and matrices.

1 Definition and Most Important Subsets of a Group.- 1. Definition of a Group.- 2. Subgroups. Normal Subgroups.- 3. The Center. The Commutator Subgroup.- 2 Homorphisms.- 4. Homomorphisms and Factors.- 5. Endomorphisms. Automorphisms.- 6. Extensions by Means of Automorphisms.- 3 Abelian Groups.- 7. Free Abelian Groups. Rank.- 8. Finitely Generated Abelian Groups.- 9. Divisible Abelian Groups.- 10. Periodic Abelian Groups.- 4 Finite Groups.- 11. Sylow p-Subgroups.- 12. Finite Simple Groups.- 13. Permutation Groups.- 5 Free Groups and Varieties.- 14. Free Groups.- 15. Varieties.- 6 Nilpotent Groups.- 16. General Properties and Examples.- 17. The Most Important Subclasses.- 18. Generalizations of Nilpotency.- 7 Soluble Groups.- 19. General Properties and Examples.- 20. Finite Soluble Groups.- 21. Soluble Matrix Groups.- 22. Generalizations of Solubility.- Append.- Auxiliary Results from Algebra, Logic and Number Theory.- 23. On Nilpotent Algebras.- 23.1. Nilpotence of Associative and Lie Algebras.- 23.2. Non-Nilpotent Nilalgebras.- 24. Local Theorems of Logic.- 24.1. Algebraic Systems.- 24.2. The Language of the Predicate Calculus.- 24.3. The Local Theorems.- 25. On Algebraic Integers.- Index of Notations for Classical Objects.