Discrete subgroup

In mathematics, a discrete subgroup of a topological group G is a subgroup H such that there is an open cover of H in which every open subset contains exactly one element of H; in other words, the subspace topology of H in G is the discrete topology. For example, the integers, Z, form a discrete subgroup of the reals, R (with the standard metric topology), but the rational numbers, Q, do not. A discrete group is a topological group G equipped with the discrete topology.

Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups. Hence, there is an 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.

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