On products of Groups with abelian subgroups of small index

TL;DR

This paper proves that groups formed by products of two subgroups, each either abelian or with a quasicyclic subgroup of index 2, are soluble with derived length at most 3, answering a specific question in group theory.

Contribution

It establishes the solubility and bounded derived length of a new class of groups formed by products of certain subgroups, extending previous results.

Findings

Groups of the form G=AB with specified subgroups are soluble.

Derived length of such groups is at most 3.

Provides a positive answer to a question in the Kourovka notebook.

Abstract

It is proved that every group of the form $G=AB$ with two subgroups $A$ and $B$ each of which is either abelian or has a quasicyclic subgroup of index $2$ is soluble of derived length at most $3$. In particular, if $A$ is abelian and $B$ is a locally quaternion group, this gives a positive answer to Question 18.95 of "Kourovka notebook" posed by A.I.Sozutov.