Finding direct product decompositions in polynomial time

A polynomial-time algorithm is produced which, given generators for a group of permutations on a finite set, returns a direct product decomposition of the group into directly indecomposable subgroups. The process uses bilinear maps and commutative rings to characterize direct products of p-groups of class 2 and reduces general groups to p-groups using group varieties. The methods apply to quotients of permutation groups and operator groups as well.