Triple factorizations by abelian groups (Q689719)

It is a well-known theorem of Kegel that if a finite group \(G\) is expressible as a triple product \(G = AB = BC = CA\) with \(A\), \(B\) and \(C\) nilpotent groups, then \(G\) is nilpotent. Now this result has been shown to be false for infinite groups, even when \(A\), \(B\), \(C\) are abelian, by Holt and Howlett, Sysak and Zaičev. Their examples have led some researchers to conclude that there is no vestige of nilpotence in a triple product of abelian groups. Here we show that this is not the case. Theorem 1: If \(G\) is a triple product of abelian groups, then every chief factor of \(G\) is central, i.e. \(G\) is a \(\overline{Z}\)-group. This result is an easy consequence of Theorem 2: Let \(G = AB\) where \(A\) and \(B\) are abelian groups. Then a chief factor of \(G\) is centralized by either \(A\) or \(B\). It seems remarkable that this simple statement about a product of abelian groups has escaped notice during the more than 35 years that have elapsed since the publication of Itô's celebrated theorem that a product of two abelian groups is metabelian. It is also shown that if \(G\) has a triple factorization by abelian groups and the abelianization of \(G\) has finite torsion-free rank, then \(G\) is locally nilpotent. The proofs use ring- theoretic techniques due to Howlett and Sysak.