On triple factorisations of finite groups

TL;DR

This paper develops a framework for understanding triple factorisations of finite groups, introducing nondegeneracy conditions, bounds on group order, and reduction techniques involving primitive actions and wreath products.

Contribution

It introduces a general framework for triple factorisations of finite groups, including nondegeneracy criteria, bounds, and reduction methods to primitive actions.

Findings

Established an upper bound for group order using subset movement results.

Identified conditions under which triple factorisations are degenerate or nondegenerate.

Developed a reduction approach to study nondegenerate cases via primitive group actions.

Abstract

This paper introduces and develops a general framework for studying triple factorisations of the form $G=ABA$ of finite groups $G$, with $A$ and $B$ subgroups of $G$. We call such a factorisation nondegenerate if $G\neq AB$. Consideration of the action of $G$ by right multiplication on the right cosets of $B$ leads to a nontrivial upper bound for $|G|$ by applying results about subsets of restricted movement. For $A<C<G$ and $B<D<G$ the factorisation $G=CDC$ may be degenerate even if $G=ABA$ is nondegenerate. Similarly forming quotients may lead to degenerate triple factorisations. A rationale is given for reducing the study of nondegenerate triple factorisations to those in which $G$ acts faithfully and primitively on the cosets of $A$. This involves study of a wreath product construction for triple factorisations.