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, 
is a discrete group, realized as a subgroup of the special linear group 
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.
