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Any group can be given the discrete topology. Since every map from a discrete space is continuous, the topological homomorphisms of a discrete group are exactly the group homomorphisms of the underlying group. Hence, there is an isomorphism between the categories of groups and of discrete groups and indeed, discrete groups can generally be identified with the underlying (non-topological) groups. With this in mind, the term discrete group theory is used to refer to the study of groups without topological structure, in contradistinction to topological or Lie group theory. It is divided, logically but also technically, into finite group theory, and infinite group theory .

If G is a finite or countably infinite group, then the discrete topology suffices to make it a zero-dimensional Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical ana. Since the only Hausdorff topology on a finite set is the discrete one, a finite Hausdorff topological group must necessarily be discrete.

There are some occasions when a topological group or Lie groupIn mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical ana is usefully endowed with the discrete topology, 'against nature'. This happens for example in the theory of the Bohr compactificationIn mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G''. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the, and in group cohomologyGroup theory Algebraic number theory Homological algebra In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequenc theory of Lie groups.

See also

• Lattice (group)See lattice for other meanings of this term, both within and without mathematics. In mathematics, a lattice in R n is a discrete subgroup of R n which spans the real vector space R n''. Every lattice in R n can be generated from a basis for the vector spa
• Frieze groupA frieze group is an infinite discrete symmetry group for a pattern on a strip (infinitely wide rectangle). The elements of the group are therefore isometries of the plane, or of a strip. As with Wallpaper groups, Frieze groups are often visualised by thi
• Wallpaper groupPlane crystallographic groups or wallpaper groups There are seventeen different types of wallpaper patterns. As opposed to the frieze patterns, these patterns cover the entire plane and can be extended infinitely in any direction on the plane. Discrete fr
• Space group
• Crystallographic group
• Crystallographic point group
• Fuchsian group
  • Modular group
• Arithmetic group
• Congruence subgroup.
• Geometric group theory
• Computational group theory