Maximal compact normal subgroups

Abstract

The main concern is the existence of a maximal compact normal subgroup K in a locally compact group G, and whether or not G/K is a Lie group. G has a maximal compact subgroup if and only if G/Go has. Maximal compact subgroups of totally disconnected groups are open. If the bounded part of G is compactly generated, then G has a maximal compact normal subgroup K and if B(G) is open, then G/K is Lie. Generalized FCgroups, compactly generated type I IN-groups, and Moore groups share the same properties. Introduction. We shall be concerned with the conditions under which a locally compact group G possesses a maximal compact normal subgroup K and whether or not G/K is a Lie group. It is known that locally compact connected groups and compactly generated Abelian groups have maximal compact normal subgroups with Lie factor groups. Moreover, almost connected groups (i.e., when G/Go is compact, where Go is the identity component of G) and compactly generated FCgroups share these results. For general locally compact Abelian groups, the corresponding assertion is not true even if the group is a-compact. Consider any Q,a of a-adic numbes; Q7a is a locally compact a-compact Abelian group which has no maximal compact (normal) subgroup. In this paper we show that a locally compact group G has a maximal compact subgroup if and only if G/Go has (Theorem 1). This generalizes substantially the corresponding result for almost connected groups mentioned before. It also reduces the question of the existence of maximal compact subgroups to that of totally disconnected groups-in which we show every maximal compact subgroup is open (Corollary 1 of Theorem 2). In [1] it is proved that a generalized FCgroup G has a maximal compact normal subgroup K. Here we show that G/K is a Lie group (Theorem 5). We also prove that, if the bounded part of G, B(G), is compactly generated, then G has a maximal compact normal subgroup (Theorem 6). Particularly, we see that compactly generated Moore groups, type I IN-groups, and compact extensions of compactly generated nilpotent groups have the desired properties (Theorems 7 and 8). Finally some sufficient conditions for G/K to be a Lie group are summarized in Theorem 10. If a topological group G has a maximal compact subgroup N, then clearly gNgis a maximal compact subgroup of G for any g C G. A maximal compact subgroup contains every compact normal subgroup. Hence K = ngeG gNg-1 is the maximal compact normal subgroup of G. Received by the editors July 12, 1985 and, in revised form, December 24, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 22D05. (?1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page