Editing Lie group (section)

=== Related concepts ===
A '''[[complex Lie group]]''' is defined in the same way using [[complex manifold]]s rather than real ones (example: <math>\operatorname{SL}(2, \mathbb{C})</math>), and holomorphic maps. Similarly, using an alternate [[Complete metric space#Completion|metric completion]] of {{tmath|1= \mathbb{Q} }}, one can define a '''''p''-adic Lie group''' over the [[p-adic number|''p''-adic numbers]], a topological group which is also an analytic ''p''-adic manifold, such that the group operations are analytic. In particular, each point has a ''p''-adic neighborhood.

[[Hilbert's fifth problem]] asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, [[Andrew Gleason|Gleason]], [[Deane Montgomery|Montgomery]] and [[Leo Zippin|Zippin]] showed that if ''G'' is a topological manifold with continuous group operations, then there exists exactly one analytic structure on ''G'' which turns it into a Lie group (see also [[Hilbert–Smith conjecture]]). If the underlying manifold is allowed to be infinite-dimensional (for example, a [[Hilbert manifold]]), then one arrives at the notion of an infinite-dimensional Lie group.  It is possible to define analogues of many [[group of Lie type|Lie groups over finite fields]], and these give most of the examples of [[finite simple group]]s.

The language of [[category theory]] provides a concise definition for Lie groups: a Lie group is a [[group object]] in the [[category (mathematics)|category]] of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to [[supergroup (physics)|Lie supergroups]]. This categorical point of view leads also to a different generalization of Lie groups, namely [[Lie groupoid|Lie groupoids]], which are [[Groupoid object|groupoid objects]] in the category of smooth manifolds with a further requirement.