On Triple Factorisations of Finite Groups

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 6= 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. 2000 Mathematics subject classification: 20B05 (primary), 20B15, 20D40 (secondary)