Computing Intersections and Normalizers in Soluble Groups

Let H and K be arbitrary subgroups of a finite soluble group G. The purpose of this paper is todescribe algorithms for constructing H@?K and N"G(H). The first author has previously described algorithms for constructing H@?K when the indices |G:H| and |G:K| are coprime, and for constructing N"G(H) when |G:H| and |H| are coprime (i.e. when H is a Hall subgroup of G). The intersection and normalizer algorithms described in the present paper are constructed from generalizations of these algorithms and from an orbit-stabilizer algorithm.