A splitting theorem and the principal ideal theorem for some infinitely generated groups

Abstract

The first theorenm of this paper is an extension of a splitting theorem for finite groups (cf. [61]) to include periodic groups certain of whose Sylow subgroups contain no elements of infinite height. The second theorem is an extension of the principal ideal theorem for finitely generated groups (cf. [71) to include all groups except again that some of the Sylow subgroups contain no elements of infinite height. A counter example will show that this latter restriction is necessary for both theorems. In the first draft of the paper the second theorem was stated for periodic groups, the proof being based onl the first theorem. The present more general statement anld proof independent of Theorem-i 1 are due to the referee to whom I am also indebted for a simplification in the proof of the lemma below.