In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup H of a topological group G is a subgroup whose induced topology is the discrete one. For example, the integers, Z, form a discrete subgroup of the reals, R, but the rational numbers, Q, do not.
Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group. Hence, there is an isomorphism between the categories of groups and of discrete groups and indeed, discrete groups can generally be identified with the underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory .
If G is a finite or countably infinite group, then the discrete topology suffices to make it a zero-dimensional Lie group. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.
There are some occasions when a topological group or Lie group is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactification, and in group cohomology theory of Lie groups.
See also