Editing Discrete group (section)

==Properties==
Since topological groups are [[homogeneous space|homogeneous]], one need only look at a single point to determine if the topological group is discrete. In particular, a topological group is discrete only if the [[singleton (mathematics)|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 have this property do not regard these groups as Lie groups). The [[identity component]] of a discrete group is just the [[trivial group|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 subgroup]]s play an important role in the theory of [[covering group]]s and [[locally isomorphic groups]]. A discrete normal subgroup of a [[connected space|connected]] group ''G'' necessarily lies in the [[center (group theory)|center]] of ''G'' and is therefore [[abelian group|abelian]].

''Other properties'':
*every discrete group is [[totally disconnected]]
*every subgroup of a discrete group is discrete.
*every [[quotient group|quotient]] of a discrete group is discrete.
*the product of a finite number of discrete groups is discrete.
*a discrete group is [[compact group|compact]] if and only if it is finite.
*every discrete group is [[locally compact group|locally compact]].
*every discrete subgroup of a Hausdorff group is closed.
*every discrete subgroup of a compact Hausdorff group is finite.