Effective black-box constructive recognition of classical groups

Abstract

We describe a black-box Las Vegas algorithm to construct standard generators for classical groups defined over finite fields. We assume that the field has size at least 4 and that oracles to solve certain problems are available. Subject to these assumptions, the algorithm runs in polynomial time. A practical implementation of our algorithm is distributed with the computer algebra system Magma.

Constructing composition factors for a linear group in polynomial time

“…A Las Vegas black-box algorithm to solve the constructive recognition problem for classical groups is described in [26]. The situation for exceptional groups is more complicated.…”

“…We [35] present Las Vegas black-box algorithms to solve the problem for the remaining exceptional groups. The algorithms of [26,35] take as input a representation of G(q) and run in time polynomial in the size of the input subject to the existence of the following oracles:…”

Recognition of finite exceptional groups of Lie type

“…More recently, Leedham-Green & O'Brien [42] developed algorithms for classical groups in natural representation and odd characteristic; those of Dietrich et al [29] apply to even characteristic. Black-box equivalents appear in [30]. All run in time polynomial in the size of the input subject to the availability of a discrete log oracle.…”

“…For all but E 8 (q) in even characteristic, our algorithms to construct the SL 2 subgroups and to label the root and toral elements are black-box provided that the algorithms employed in Theorem 2.2 for constructive recognition of small rank classical groups are black-box. Since algorithms are available for these tasks (see, for example, [30] and its references), a version of Theorem 1 could be formulated for black-box groups. We refrain from doing so.…”

Constructing All Composition Series of a Finite Group

“…The third is (possibly Monte-Carlo, possibly assuming a discrete logarithm oracle) polynomial time for most classes of simple groups groups [8,15,13], with the exception of composition factors of type 2 G2(q). One can of course dispense with these qualifiers if the size of nonabelian composition factors is bounded.…”