Discrete group

[1] A subgroup H of a topological group G is a discrete subgroup if H is discrete when endowed with the subspace topology from G. In other words there is a neighbourhood of the identity in G containing no other element of H. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not.

In particular, a topological group is discrete only if the singleton containing the identity is an open set.

Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.

A discrete subgroup H of G is cocompact if there is a compact subset K of G such that HK = G. Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups.

A discrete normal subgroup of a connected group G necessarily lies in the center of G and is therefore abelian.