A discrete group is a topological group with the discrete topology. Often in practice, discrete groups arise as discrete subgroups of continuous Lie groups acting on a geometric space. For example, SL_2(Z) is a discrete group, realized as a subgroup of the special linear group SL_2(R) acting on the upper half-plane by Möbius transformations. Discrete groups also appear naturally as symmetries of discrete structures (e.g., graphs, tilings, lattices, polyhedra), fundamental groups of topological spaces, and other related structures.

discrete group algebra | discrete group theory | fundamental group | Lie group