Intersecting subgroups of finite soluble groups

• Authors: S. P. Glasby
• Venue: Journal of Symbolic Computation
• Year: 1988

This paper describes an algorithm for constructing certain important subgroup intersections H @? K, of a finite soluble group G. The algorithm is applicable if the index of H in G is coprime to the index of K in G, or more generally, if each hypereccentric factor of G is covered by H or K. If G has composition length n, then the intersection algorithm computes H @? K using O(n^3 log|G|) group multiplications. This intersection algorithm can be applied to compute Sylow bases, system normalisers and complements to abelian terms of the lower nilpotent series. These structures are particularly useful for constructing the automorphism group of G. Intersecting arbitrary subgroups of a finite soluble group is, in general, a difficult problem. This is shown by an example where the subgroups H and K of G both avoid an eccentric factor of G, and computing H @? K is equivalent to constructing a discrete logarithm in a finite field. However, there are no known polynomial algorithms for constructing discrete logarithms.