A constructive recognition algorithm for the special linear group

Celler, Frank; Leedham‐Green, C. R.

doi:10.1017/cbo9780511565830.007

Cited by 16 publications

(22 citation statements)

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“…Of course, when our results are used elsewhere the classification will , presumably, be needed in order to obtain our hypotheses. Nevertheless, the present approach, and those of [CFL,CLG2], have the advantage of dispensing with difficult group theory and aiming more directly at the interesting algorithmic aspects of questions such as these.…”

Section: Is Yet Another Corollary: a Monte Carlo Algorithm That Decidmentioning

confidence: 99%

“…The algorithm in [CFL] leads to better timing estimates than the one presented here in the case PSL(d, 2) studied in that paper. As in [CFL], we cannot use Gaussian elimination as was done in [CLG2]: that reference already had the "correct" vector space within which to compute. On the other hand , Gaussian elimination enabled [CLG2] to contain a more efficient and practical algorithm, which has already been implemented in GAP and MAGMA.…”

Section: Is Yet Another Corollary: a Monte Carlo Algorithm That Decidmentioning

confidence: 99%

“…At present, the main examples of this need occur in finding Sylow subgroups of permutation groups [Ka3,Ma,KLM,Mol,M03] and in recognition algorithms for quasisimple matrix groups [CLG2,Ce]. Whereas the preceding references deal with special instances of this type of question, a recent breakthrough by Cooperman, Finkelstein and Linton [CFL] studies a much more general setting.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Black box classical groups

Kantor¹,

Seress²

2001

Memoirs of the AMS

167

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Section: Is Yet Another Corollary: a Monte Carlo Algorithm That Decidmentioning

confidence: 99%

Section: Is Yet Another Corollary: a Monte Carlo Algorithm That Decidmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Black box classical groups

Kantor¹,

Seress²

2001

Memoirs of the AMS

167

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“…The black box recognition algorithms for classical groups proposed by Kantor & Seress [22] and the recognition algorithm for SL(d, q) in its natural representation proposed by Celler & Leedham-Green [10] have complexity that is polynomial in the size of the field (and the rank of the group). Classical groups have faithful representations over the defining field (such as the natural representation) whose degrees are independent of the field size, and so the field size may be exponential in the size of the input.…”

Section: Introductionmentioning

confidence: 99%

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Conder

Leedham‐Green

O’Brien

2005

Trans. Amer. Math. Soc.

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Abstract. Existing black box and other algorithms for explicitly recognising groups of Lie type over GF(q) have asymptotic running times which are polynomial in q, whereas the input size involves only log q. This has represented a serious obstruction to the efficient recognition of such groups.Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises PSL(2, q) explicitly.The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising SL(2, q) in its natural representation in polynomial time, given a discrete logarithm oracle for GF(q). The algorithm presented here takes as input a generating set for a subgroup G of GL(d, F ) that is isomorphic modulo scalars to PSL(2, q), where F is a finite field of the same characteristic as GF(q); it returns the natural representation of G modulo scalars. Since a faithful projective representation of PSL(2, q) in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in q rather than in log q, elementary algorithms will recognise PSL(2, q) explicitly in polynomial time in these cases. Given a discrete logarithm oracle for GF(q), our algorithm thus provides the required polynomial time oracle for recognising PSL(2, q) explicitly in the remaining case, namely for representations in the natural characteristic.This leads to a partial solution of a question posed by Babai and Shalev: if

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“…However, the algorithm in [CLG2] producing X * does not quite run in polynomial time: there is a factor q in the timing, where V is a vector space over F q . The corresponding result has also been proved for all classical groups: given G = X ≤ GL(d, q) having a normal classical subgroup C defined on V , algorithms in [Ce,Bro1,Bro2] output, with high probability, a new generating set X * such that there is a polynomial-time procedure that gets from X * to any given g ∈ G. The version of this theorem in [Bro2] handles all symplectic, orthogonal and unitary groups simultaneously in a more or less uniform manner.…”

Section: Constructive Recognition Of Simple Groupsmentioning

confidence: 99%

Computing with matrix groups

Kantor¹

2003

Groups, Combinatorics and Geometry

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A constructive recognition algorithm for the special linear group

Cited by 16 publications

References 0 publications

Black box classical groups

Black box classical groups

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Computing with matrix groups

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