Algorithms for matrix groups

Abstract

Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F ), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G.

Recognizing finite matrix groups over infinite fields

“…299-300] that requires a presentation of φ ρ (G) as input. Since it is a matrix group over a finite field, we can compute a presentation of φ ρ (G) using the algorithms described in [1,21]. We refer to such an algorithm as Presentation.…”

“…The algorithms have been implemented in MAGMA as part of our package INFINITE [10]. We use machinery from the COMPOSITIONTREE package [1,21] to study congruence images and construct their presentations.…”

“…If a group G is finite then, in practice, we can often construct an isomorphic copy of G over some finite field. As a consequence, drawing on recent progress in computing with matrix groups over finite fields [1,21], we obtain the first algorithms to answer many structural questions about G. These include: computing |G|; testing membership in G; computing Sylow subgroups, a composition series, and the solvable and unipotent radicals of G.…”

Algorithms for the Tits alternative and related problems

“…, Ψ(g r ). Such presentations can be computed using algorithms from [4,22]. The relators in P are then evaluated by replacing each occurrence of Ψ(g i ) in each relator by g i , 1 ≤ i ≤ r. The resulting words in the g i constitute the output of NormalGenerators.…”

Effective black-box constructive recognition of classical groups

“…For an overview of the Matrix Group Recognition Project, to which this work contributes, see [37]. Much of the background and preliminaries needed for this paper are summarised in [19,26,37].…”