%0 Journal Article
%T On triple factorisations of finite groups
%A S. Hassan Alavi
%A Cheryl E. Praeger
%J Mathematics
%D 2009
%I arXiv
%X 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.
%U http://arxiv.org/abs/0909.4393v1