On the Structure of Certain Locally Compact Topological Groups

A locally compact topological group G is called an (H) group if G has a maximal compact normal subgroup with Lie factor. In this note, we study the problem when a locally compact group is an (H) group. Let G be a locally compact Hausdorff topological group. Let G0 be the identity component of G. If G/G0 is compact, then we say that G is almost connected. The structure of almost connected locally compact groups is well understood and it is the consequence of the solution of the Hubert fifth problem. Roughly speaking, the structure of almost connected locally compact groups can be described by the following statement: Every almost connected locally compact group has small normal subgroups such that the quotient groups are Lie groups and the group itself is the inverse limit of these Lie groups in a natural way. We cannot expect locally compact groups in general to have such properties, so we ask when a locally compact group has a compact normal subgroup such that the quotient group is a Lie group. The purpose of the present note is to provide some partial answers to this question. Now, let G be a locally compact group. We may view G as an extension of G0 by a totally disconnected locally compact topological group G/G0 . Since the natures of these two types of groups are very different, it is convenient to study them separately first and then put results together afterwards. Because our knowledge of connected locally compact groups is by far superior to what we know about totally disconnected locally compact groups, our main effort lies on totally disconnected locally compact groups. Actually, we do not lose any generality approaching our problem by this method. If G/GQ has a compact normal subgroup such that the quotient group is a Lie group (i.e., a discrete group), then G has an open normal subgroup Gx suchthat Gx/G0 is compact. Then (7, has a maximal compact normal subgroup K so that G, ¡K is a Lie group, since C7, is almost connected. Since K is characteristic in Gx, it is normal in G. Therefore G/K is a Lie group. Notice that every totally disconnected locally compact group has small compact open subgroups. Frequently, Received by the editors July 26, 1988 and, in revised form, May 3, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 22A05. © 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page