The University, Manchester. GROUPS COVERED BY PERMUTABLE SUBSETS

In this paper we shall be mainly concerned with groups which can be covered by (in other words, are unions of) permutable boundedly finite subsets. Obvious examples of such groups are the finite groups, where a single finite set suffices, namely the group itself; and abelian groups, where all one-element subsets will serve. The main result (Theorem 7.1) characterizes these groups completely as those groups which possess a subgroup of finite index with finite derived group. Another, closely related, class of groups, which also includes all finite groups and all abelian groups, is that of the groups with only finite classes of conjugate elements. Such groups are called jFC-groupsf; they have been studied in an earlier paper J, and their investigation is carried a little further in the present paper. I t will be shown (Theorem 3.1) that if the classes of conjugate elements in a group H are boundedly finite, then the derived group of H is finite. The converse is also true, and nearly trivial. The present investigation arose from the question whether a simple and direct proof of the following theorem of Mautner§ could be devised: