Constructive recognition of a black box group isomorphic to GL(n,2)

Abstract

A Monte Carlo algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n, 2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data structure used to compute the isomorphism and each invocation of the isomorphism require polynomial time complexity. The ultimate goal is to eventually extend this result to the larger question of constructing the natural representation of classical groups. Extensions of the methods developed in this paper are discussed as well as open questions.