Minimal normal systems of compact right topological groups

Abstract

We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.