Editing Discrete group (section)

{{short description|Type of topological group}}
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[[File:Number-line.svg|right|thumb|300px|The integers with their usual topology are a discrete subgroup of the real numbers.]]

In [[mathematics]], a [[topological group]] ''G'' is called a '''discrete group''' if there is no [[limit point]] in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and only if its [[Identity element|identity]] is [[Isolated point|isolated]].{{sfn|Pontrjagin|1946|p=54}} 

A [[subgroup]] ''H'' of a topological group ''G'' is a '''discrete subgroup''' if ''H'' is discrete when endowed with the [[induced topology|subspace topology]] from ''G''. In other words there is a neighbourhood of the identity in ''G'' containing no other element of ''H''. For example, the [[integer]]s, '''Z''', form a discrete subgroup of the [[real number|reals]], '''R''' (with the standard [[Metric space|metric topology]]), but the [[rational number]]s, '''Q''', do not. 

Any group can be endowed with the [[discrete topology]], making it a discrete topological group. Since every map from a discrete space is [[Continuous (topology)|continuous]], the topological homomorphisms between discrete groups are exactly the [[group homomorphism]]s between the underlying groups. Hence, there is an [[Isomorphism of categories|isomorphism]] between the [[category of groups]] and the category of discrete groups. Discrete groups can therefore be identified with their underlying (non-topological) groups.

There are some occasions when a [[topological group]] or [[Lie group]] is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the [[Bohr compactification]], and in [[group cohomology]] theory of Lie groups.

A discrete [[isometry group]] is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a [[discrete set]]. A discrete [[symmetry group]] is a symmetry group that is a discrete isometry group.