Discrete Group - Properties

Properties

Since topological groups are homogeneous, one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete if and only if the singleton containing the identity is an open set.

A discrete group is the same thing as a zero-dimensional Lie group (uncountable discrete groups are not second-countable so authors who require Lie groups to satisfy this axiom do not regard these groups as Lie groups). The identity component of a discrete group is just the trivial subgroup while the group of components is isomorphic to the group itself.

Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete. It follows that every finite subgroup of a Hausdorff group is 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.

Other properties:

• every discrete group is totally disconnected
• every subgroup of a discrete group is discrete.
• every quotient of a discrete group is discrete.
• the product of a finite number of discrete groups is discrete.
• a discrete group is compact if and only if it is finite.
• every discrete group is locally compact.
• every discrete subgroup of a Hausdorff group is closed.
• every discrete subgroup of a compact Hausdorff group is finite.