Examples of Groups: Infinite Discrete Groups

Abstract

We proceed now to consider infinite groups. Of course, the purely negative characteristic of not being finite does not reflect the situations which really arise. Usually the infinite set of elements of a group is defined by some constructive process or formula. This formula may contain some parameters, which may take integer values, or be real numbers, or even points of a manifold. This is the starting point of an informal classification: groups are called discrete in the first case, and continuous in the second. The simplest example of a discrete group is the infinite cyclic group, whose elements are of the form g ⁿ where n runs through all the integers.