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.
Read more about this topic: Discrete Group
Famous quotes containing the word properties:
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—John Locke (16321704)
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—Ralph Waldo Emerson (18031882)