An operator \(A\) acting on a Banach space \(X\) is called generalised Fredholm operator if there exists an operator \(S\) on \(X\) such that \(ASA=A\) and \(I-SA-AS\) is a Fredholm operator. These operators have been studied by the author in several previous papers. Let \(A\) and \(B\) be operators on \(X\) and let \(\Phi_g(X)\) denote the space of generalised Fredholm operators on \(X\). The main results in this paper establish that \(AB\) Fredholm implies that \(BA\in \Phi_g(X)\), and that \((AB)^n\in \Phi_g(X)\) implies that \((BA)^{n+1}\in \Phi_g(X)\); in particular, \(A^n\in \Phi_g(X)\) implies that \(A^{n+1}\in \Phi_g(X)\).