Maximal compact subgroup

of a topological group

A compact subgroup (cf. Compact group) which is not contained as a proper subgroup in any compact subgroup of . For example, for , for a solvable simply-connected Lie group .

In an arbitrary group maximal compact subgroups need not exist (for example, if , where 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 is a connected Lie group, then any compact subgroup of is contained in some maximal compact subgroup (in particular, maximal compact subgroups must exist) and all maximal compact subgroups of are connected and conjugate to each other. The space of the group is diffeomorphic to , therefore most of the topological questions about Lie groups reduce to the corresponding questions for compact Lie groups (cf. Lie group, compact).

References

[1]	E. Cartan, "La géometrie des groupes de transformations" J. Math. Pures Appl. , 6 (1927) pp. 1–119
[2]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)