A non-constructive recognition algorithm for the special linear and other classical groups

Black box classical groups

“…Aschbacher [As1] classified the subgroups of general linear groups into several categories, and recent practical computations with matrix groups center around the problem of finding a category to which a given matrix group belongs [NP,NiP1,NiP2 ,CLG1 ,CLG2,HR1 ,HR2 ,HLOR1 ,HLOR2, LGOj. One of Aschbacher's categories is that the subgroup G of GL(n,pC) has a normal quasisimple classical matrix group, in its natural representation.…”

Computing with matrix groups

“…This theorem was followed by others [NiP,CLG1] that decide, similarly, whether or not a given subgroup G = X ≤ GL(d, q) contains a classical group defined on V as a normal subgroup.…”

Black-box recognition of finite simple groups of Lie type by statistics of element orders

“…Some of these [13,25,26] take a matrix group G = S ≤ GL(d, q) as input, and decide by a polynomial-time one-sided Monte Carlo algorithm whether G is a classical group defined on the d-dimensional vector space over GF(q). (We refer to Section 2 for the definition of Monte Carlo algorithms.)…”

“…In contrast to other recent Monte Carlo recognition algorithms for classical groups [13,25,26] mentioned above, we do not even start with knowledge of the correct dimension or field, thereby enhancing the possibilities for applications of our results (e.g., in [5,24]). As with other algorithmic investigations into groups of Lie type, not having linear algebra available has required entirely different types of methodologies to be developed.…”