On products of groups which contain Abelian subgroups of index at most 2. (Q2842802)

A famous theorem of N. Itô states that if a group \(G=AB\) is the product of two Abelian subgroups \(A\) and \(B\), then \(G\) is metabelian. Itô's theorem is the first and one of the few results concerning arbitrary groups admitting a factorization by two subgroups with a given property. Much more recently, \textit{B. Amberg, A. Fransman} and \textit{L. Kazarin} [J. Algebra 350, No. 1, 308-317 (2012; Zbl 1258.20030)] have proved that any product \(G=AB\) of two periodic locally dihedral subgroups is soluble.NEWLINENEWLINE In the paper under review, the authors improve this latter result, by showing that if a group \(G=AB\) is a product of two subgroups \(A\) and \(B\), each of which is either Abelian or generalized dihedral, then \(G\) is soluble. Here a group \(X\) is called generalized dihedral if it contains an Abelian subgroup \(Y\) of index \(2\) and an involution \(x\in X\setminus Y\) inverting every element of \(Y\).