Disjoint direct product decomposition of permutation groups

Abstract

Let $H \leq S_n$ be an intransitive group with orbits $\Omega_1, \Omega_2, \ldots ,\Omega_k$. Then certainly $H$ is a subdirect product of the direct product of its projections on each orbit, $H|_{\Omega_1} \times H|_{\Omega_2} \times \ldots \times H|_{\Omega_k}$. Here we provide a polynomial time algorithm for computing the finest partition $P$ of the $H$-orbits such that $H = \prod_{c \in P} H|_c$ and demonstrate its usefulness in some applications.