Constructive homomorphisms for classical groups

Let Ω ≤ GL(V ) be a quasisimple classical group in its natural representation over a finite vector space V , and let ∆ = NGL(V )(Ω). We construct the projection from ∆ to ∆/Ω and provide fast, polynomial-time algorithms for computing the image of an element. Given a discrete logarithm oracle, we also represent ∆/Ω as a group with at most 3 generators and 6 relations. We then compute canonical representatives for the cosets of Ω. A key ingredient of our algorithms is a new, asymptotically fast method for constructing isometries between spaces with forms. Our results are useful for the matrix group recognition project, can be used to solve element conjugacy problems, and can improve algorithms to construct maximal subgroups.