A practical model for computation with matrix groups

Effective black-box constructive recognition of classical groups

“…In practical applications, when we use the methods of CompositionTree [7], we work with groups having classical groups as homomorphic images and construct kernels to these homomorphisms; now a small fixed number of standard generators is useful. Table 1 lists the principal contributors to the stated complexity of each of the algorithms of [11][12][13][14] and also the comparable costs of our algorithm.…”

“…Now consider the cost of χ, the oracle to recognise a central quotient of SL 2 (q). The algorithm of [25] produces inverse isomorphisms between a black-box copy of SL 2 (2 e ) and the natural copy in time that is polynomial in e. A similar algorithm appears in [8] for q ≡ 1 mod 4; it is polynomial in log q and the square of the characteristic of GF(q). But the only known way of producing such an isomorphism with complexity that is polynomial in log q when the characteristic is not bounded is the algorithm of [16], which applies when the group is given as a matrix representation in the defining characteristic, and assumes a discrete logarithm oracle for GF(q).…”

“…If G ≤ GL n (F) is an absolutely irreducible representation of SX d (q), with n ≤ d 2 , and F and GF(q) have the same characteristic, then Corr's implementation of the Las Vegas algorithms of [17,32] is used to construct the projective action of G on GF(q) d ; then G can be constructively recognised by our algorithms of [19,26]. To all individual base cases, we apply (our implementations of) specially designed base-case algorithms or CompositionTree [7]. We observe that the latter also readily constructs standard generators for many representations of moderate dimension of SX d (2) for d ≤ 20.…”

Constructing All Composition Series of a Finite Group

“…Using matrix group recognition [1] there is no fundamental obstacle to apply it also to matrix groups, though implementation would be harder.…”

Recognizing finite matrix groups over infinite fields

“…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.…”