A note on compact subgroups of topological groups

Our main concern is the existence of maximal compact subgroups in a locally compact topological group. If G is a locally compact group such that P(G/G _o), the set of periodic points of G/G_o, is a compact subgroup of G/G_o, than G has maximal compact subgroups K such that G/N is a Lie group where N = ∩ K, the intersection of the collection K of all maximal compact subgroups of G. Also every compact subgroup of G is contained in a maximal compact subgroup. We given an example of a discrete group which has maximal finite subgroup and has finite subgroups not contained in maximal finite subgroups. We note that the above result on P(G/G_o) is an extension of the well-known corresponding result for almost connected groups.