在数学中，离散群是配备了离散拓扑的群 G。带有这种拓扑 G 成为了拓扑群。拓扑群 G 的离散子群是其相对拓扑为离散拓扑的子群 H。例如，整数集 Z 形成了实数集 R 的离散子群，但是有理数集 Q 不行。

任何群都可以给予离散拓扑。因为出自离散空间的所有映射都是连续的，离散群的拓扑同态完全就是底层群的群同态。因此，在群范畴和离散群范畴之间有一个同构，离散群因此同一于它们的底层(非拓扑)群。由于这个想法，术语离散群论被用来称呼对没有拓扑结构的群的研究，用来对比于拓扑群论或李群论。它在逻辑上和技术上被分为有限群论和无限群论。

In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of H in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology. 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. A discrete group is a topological group G equipped with the discrete topology.