Normal subgroups of groups which are products of two Abelian subgroups

Abstract

Abstract.It is shown that if a group G=AB, where A and B are Abelian subgroups of G, A^B, and either A or B satisfies the maximum condition, then there is a normal subgroup N of G, N^G, such that N contains either A or B.Summary and background.The purpose of this paper is to show that, if a group G is a nontrivial product of two Abelian subgroups A and B, and if either A or B satisfies the maximum condition, then there is a proper normal subgroup of G containing either A or B. (L. E. Knop [7]).The basic impetus for this research is found in a paper of N. Itô [1]. In this paper, Itô showed that if G is a product of two Abelian subgroups A and B, then G is metabelian. Furthermore, he showed that if G is finite, then there is a normal subgroup of G, not the identity, which is contained in either A or B, and also that there is a proper normal subgroup of G which contains either A or B.Additional work on the problem of when there is a normal subgroup contained in one of the two factors has been done by P. M. Cohn