On the Structure of Certain Factorizable Groups, I

1. A considerable amount is known about the structure of finite factorizable groups—that is, groups G which can be represented in the form AB, where A and B are subgroups of G. Such groups are known to be solvable under a variety of assumptions on the subgroups A and B. If A and B are both Abelian, Ito [6] has shown that G is actually metabelian. If A and B are both cyclic, it is easy to see that either a subgroup of A or a subgroup of B must be normal in G (Douglas [l]). Our investigations into the structure of finite groups of the form ABA, A, B being cyclic, [4; 5] have included AB groups as a special case. For our further work on this subject, it has been necessary to determine the precise structure of an AB group in which A is its own normalizes Our results are contained in the following theorem.