Editing Group theory (section)

=== Groups with additional structure ===
An important elaboration of the concept of a group occurs if ''G'' is endowed with additional structure, notably, of a [[topological space]], [[differentiable manifold]], or [[algebraic variety]]. If the multiplication and inversion of the group are compatible with this structure, that is, they are [[continuous map|continuous]], [[smooth map|smooth]] or [[Regular map (algebraic geometry)|regular]] (in the sense of algebraic geometry) maps, then ''G'' is a [[topological group]], a [[Lie group]], or an [[algebraic group]].<ref>This process of imposing extra structure has been formalized through the notion of a  [[group object]] in a suitable [[category (mathematics)|category]]. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.</ref>

The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for [[abstract harmonic analysis]], whereas [[Lie group]]s (frequently realized as transformation groups) are the mainstays of [[differential geometry]] and unitary [[representation theory]]. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, [[compact Lie group|compact connected Lie groups]] have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group ''Γ'' can be realized as a [[lattice (discrete subgroup)|lattice]] in a topological group ''G'', the geometry and analysis pertaining to ''G'' yield important results about ''Γ''. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups ([[profinite group]]s): for example, a single [[powerful p-group|''p''-adic analytic group]] ''G'' has a family of quotients which are finite [[p-group|''p''-groups]] of various orders, and properties of ''G'' translate into the properties of its finite quotients.