From Wikipedia, the free encyclopedia.
In mathematics, a discrete group is a group G together with the discrete topology. With this topology G becomes a topological group. Since any group can be given the discrete topology any group can be regarded as a discrete topological group. If G is finite or countable then it is also 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.
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.
