A remark on double cosets

TL;DR

This paper proves that a soluble group with two finitely generated abelian subgroups having finitely many double cosets must be virtually polycyclic, revealing a structural property related to double coset finiteness.

Contribution

It establishes a new criterion linking double coset finiteness of abelian subgroups to the virtual polycyclicity of soluble groups.

Findings

Finitely generated abelian subgroups with finitely many double cosets imply the group is virtually polycyclic.

Provides a structural characterization of soluble groups based on double coset properties.

Abstract

If a soluble group $G$ contains two finitely generated abelian subgroups $A,B$ such that the number of double cosets $AgB$ is finite, then $G$ is shown to be virtually polycyclic.