On normal subgroups of factorized groups (Q1179846)

A well-known result of Itô says that a non-trivial finite group \(G=AB\) which is the product of two abelian subgroups \(A\) and \(B\) always has a non-trivial normal subgroup contained in \(A\) or \(B\). Here it is proved that for the existence of such a normal subgroup of \(G=AB\) it suffices that \(G\) has a non-trivial normal minimax subgroup. It is also shown that an infinite nilpotent-by-abelian group \(G=AB\) of finite abelian section rank, which is the product of two central-by-finite subgroups \(A\) and \(B\), has an infinite normal subgroup contained in \(A\) or \(B\). In the proofs the following theorem on triple factorized groups is needed: If the group \(G=AB=AK=BK\) with finite abelian section rank is the product of three finite-by-nilpotent subgroups \(A\), \(B\) and \(K\), where \(K\) is normal in \(G\), then the hypercentre of \(G\) has finite index in \(G\).