On compactification and structure of topological groups

Abstract

The class of locally compact groups G with center Z such that G/Z is compact was investigated by Grosser and Moskowitz [5 ], [6 ]. They show that these groups have equal left and right uniformities, that they are maximally almost periodic, and that they are of the forml VXH where V is a vector group, and H has a compact open normal subgroup. It is well known that a group is maximally almost periodic if and only if it has a compactification in the sense that there is a continuous isomorphism of the group into a compact group [2]. For locally compact connected groups G, the following are equivalent; (1) G/Z is conmpact, (2) G has a compactification and (3) G has equal uniformities [14]. In this paper we are primarily interested in finding other classes of groups for which the three properties mentioned above are related. For example, in Theorem 6, we show that (1), (2) and (3) are equivalent for a certain class of compactly generated groups. The following lemma is a natural topological version of a group theoretic construction. For a group G, Z = Z(G) is the center of G.