A note on products of abelian by finite groups (Q912211)

A famous theorem of \textit{N. Itô} [Math. Z. 62, 400-401 (1955; Zbl 0064.25203)] states that every product of two abelian groups is metabelian. It was later shown by \textit{N. S. Chernikov} [Ukr. Mat. Zh. 33, 136-138 (1981; Zbl 0466.20008)] that a group \(G=AB\), which is factorized by two central-by-finite subgroups \(A\) and \(B\), is soluble-by-finite. It is an open question whether every product of two abelian-by-finite groups is metabelian-by-finite. In the paper under review, the author proves that a residually finite group \(G=AB\), which is the product of two abelian-by-finite subgroups \(A\) and \(B\), is metabelian-by-finite. The proof uses the fact that the profinite completion of \(G\) is factorized by two closed abelian-by-finite subgroups.