Conjugacy of Subgroups of the General Linear Group

In this paper we present a new, practical algorithm for solving the subgroup conjugacy problem in the general linear group. Mathematics Subject Classification: 20-04, 20H30. 1 Introductory Material This paper presents a new algorithm to solve a subcase of the following: Problem. Given two groups G,H ≤ K, determine whether there exists a k ∈ K such that G = H. If so, return one such k. This problem is known as the subgroup conjugacy problem, and is computationally difficult to solve. The usual approach is to modify algorithms for computing normalisers of subgroups, since the set of elements of K which conjugate G to H, if nonempty, is a coset of NK(G). Butler developed a backtrack search algorithm for permutation and matrix groups [4], and used this to compute normalisers of permutation groups, and to solve the subgroup conjugacy problem in permutation groups [5]. Butler’s ideas for computing subgroup normalisers were extended by Holt [12], but only for permutation groups. More ∗The author would like to thank Charles Leedham-Green, Derek Holt and John Cannon for their advice during the writing of this article. Much of this work was carried out at the University of Sydney, where I was partially supported by a grant from the Australian Research Council. I have since been supported by the EPSRC, grant number GR/S30580/01.