chapter  1
50 Pages

Groups

WithL. S. Pontryagin

This chapter presents the exposition of the fundamental concepts of group theory. The theory of groups studies an algebraic operation in its purest form: the elements constituting the group are considered only from the point of view of the operation defined in the group, all other possible properties of the elements being ignored. In the sequel we shall frequently have occasion to consider various subsets of a group, and to perform certain operations upon them. Among the subgroups of any group are those consisting of all integral powers of an arbitrary one of its elements. A group consisting exclusively of the powers of some one of its elements is said to be cyclic. The correctness of this definition is demonstrated word for word as in the case of addition.