View source for Maximal compact subgroup

{{TEX|done}}{{MSC|22A05}}

''of a [[topological group]] $G$''

A compact subgroup (cf. [[Compact group]]) $K \subset G$ which is not contained as a proper subgroup in any compact subgroup of $G$. For example, $K = \text{SO}(n)$ for $G = \text{GL}(n,\mathbf{R})$; $K=\{e\}$ for a solvable simply-connected Lie group $G$.

In an arbitrary group $G$ maximal compact subgroups need not exist (for example, if $G = \text{GL}(V)$, where $V$ is an infinite-dimensional Hilbert space), and if they do exist there may be non-isomorphic ones among them.

Maximal compact subgroups of Lie groups have been studied most. If $G$ is a connected Lie group, then any compact subgroup of $G$ is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of $G$ are connected and conjugate to each other. The space of the group $G$ is diffeomorphic to $K \times \mathbf{R}^n$, therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. [[Lie group, compact]]).

====References====
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,   "La géometrie des groupes de transformations"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 1–119  {{ZBL|53.0388.01}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  {{ZBL|0451.53038}}</TD></TR>
</table>