  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.
