On Some Types of Topological Groups

Abstract

As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure