Groups

Abstract

The concept of a group is of fundamental importance in the study of algebra. Groups which are, from the point of view of algebraic structure, essentially the same are said to be isomorphic. Ideally the goal in studying groups is to classify all groups up to isomorphism, which in practice means finding necessary and sufficient conditions for two groups to be isomorphic. At present there is little hope of classifying arbitrary groups. But it is possible to obtain complete structure theorems for various restricted classes of groups, such as cyclic groups (Section 3), finitely generated abelian groups (Section II.2), groups satisfying chain conditions (Section II.3) and finite groups of small order (Section II.6). In order to prove even these limited structure theorems, it is necessary to develop a large amount of miscellaneous information about the structure of (more or less) arbitrary groups (Sections 1, 2, 4, 5, and 8 of Chapter I and Sections 4 and 5 of Chapter II). In addition we shall study some classes of groups whose structure is known in large part and which have useful applications in other areas of mathematics, such as symmetric groups (Section 6), free [abelian] groups (Sections 9 and II.1), nilpotent and solvable groups (Sections II.7 and II.8).