J. Group Theory 4 (2001), 427±455 Journal of Group Theory ( de Gruyter 2001 Locally compact groups with many automorphisms Markus Stroppel (Communicated by S. A. Morris) Abstract. Locally compact groups whose number of orbits under the group of all topological automorphisms is bounded by some cardinal number b are studied in this paper. The assump- tion turns out to be restrictive if the group is compact or connected and b is countable, or if b W 3, or if the group is abelian and b is ®nite. 1 Introduction The purpose of the present paper is to derive structural constraints for a locally compact group G from the assumption that G has `many' automorphisms in the fol- lowing sense: the number oG of orbits of the group AutG of all (topological) automorphisms of G is ®nite. Groups G with oG W 2 were called homogeneous in [20]. In that paper, additive groups of topological vector spaces of ®nite dimension over a local ®eld were characterized as the homogeneous locally compact groups, subject to mild additional topological hypotheses (see 2.13 below for details). If one drops these additional hypotheses, however, there are locally compact abelian homogeneous groups that do not allow any continuous multiplication by scalars from any ®eld; cf. 2.11. See 2.6, 2.10 and 2.13 below for more information about homogeneous groups. We will ®nd rather explicit descriptions for G if oG W 3; that is, for the groups that were called almost homogeneous in [14] and in [21]. See 6.6 below. For a compact or connected locally compact group G, even the condition oG < 2 @ 0 turns out to be very restrictive; see 4.4 and 4.9. Organization of the paper. After a collection of illustrative examples in Section 2, we compile some basic lemmas in Section 3. The remaining sections deal with the impact of various topological or algebraic assumptions together with suitable bounds on the number of orbits. Section 4 shows that the condition oK  < @ 0 is quite restrictive for a compact group K ; see 4.6, 4.7, 4.8. The case when oK  W 4 is discussed com- pletely in 4.9. The next section contains special information on locally compact con- nected groups H with oH < 2 @ 0 : it follows from 4.4 that H is a simply connected Brought to you by | University of Arizona Authenticated Download Date | 7/6/15 10:05 PM