A study of the proximal relation in coset transformation groups

Abstract

1. Introduction. In this paper, we investigate the proximal relation and related notions in coset transformation groups. In ?1, we show that many of the interesting properties of the proximal relation can be characterized in terms of a group subset that arises naturally from the generating subgroup. A general criterion for distal in terms of subnormality of the generating subgroup is shown in ?2. Proofs are given for [9, Remark (2.1) and Theorem (2.2)], establishing that coset transformation groups of nilpotent groups are distal. ?3 is concerned with distal in relation to certain coset transformation groups that arise from solvable Lie groups. Finally, we establish in ?4 some results concerning the existence of subgroups which yield either point-transitive or minimal distal actions on the phase space. These results are strengthened when applied to certain discrete flows and examined in Lie groups. Proofs for the rest of the results of [9] are given in this section. Let G be a topological group and H a subgroup of G. Consider the space H\G = {Hg I g e G} of right cosets of G. The coset transformation group of G induced by H is defined to be the point-to-point transitive transformation group (H\G, G) with action (Hf, g) -- Hfg. The fact that (H\G, G) is indeed a transformation group under this action follows from [6, Definition 1.39]. If H is syndetic in G, H\G is compact uniformizable, whence H\G has a unique compatible uniformity. Thus, the proximal relation of (H\G, G) is well defined and will be denoted by P(H\G). Throughout this paper, G will denote a topological group with identity e and H will denote a syndetic subgroup. If K and L are subgroups of G, we denote that K is normal in L by K < L. If K is a subgroup of G, then K* will denote the closure of K. The additive groups of integers and reals will be denoted by Z and R respectively. The neighborhood filter of e will be denoted by Tie. For other definitions,