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

Babai, László; Kantor, William M.; Pálfy, P. P.; Seress, Ákos

doi:10.1515/jgth.2002.010

Cited by 28 publications

(36 citation statements)

References 23 publications

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“…The critical case is when H is a group of Lie type, with natural characteristic p. We may use the (non-constructive) polynomial-time Monte Carlo algorithm of Babai et al [1] to name the group. Given an oracle that recognises PSL(2, q) explicitly, the algorithms of Brooksbank & Kantor [6] construct Chevalley generating sets for the linear and symplectic groups in time polynomial in the size of the input.…”

Section: Deciding Isomorphismmentioning

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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Section: Deciding Isomorphismmentioning

confidence: 99%

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Conder

Leedham‐Green

O’Brien

2005

Trans. Amer. Math. Soc.

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“…The main result of [BaB] is a polynomial-time algorithm to find a black-box representation of characteristic p for all non-abelian composition factors of a black-box group of characteristic p. Combining this with recent results on the statistical recognition of finite simple groups ( [AltB], [KS2], [BaKPS]), we are now able to name the nonabelian composition factors: Theorem 1.1 ( [BaB,AltB,KS2,BaKPS]). Given a black-box group G of known characteristic, the standard names of all nonabelian composition factors of G can be computed in Monte Carlo polynomial time.…”

Section: Nonabelian Composition Factorsmentioning

confidence: 87%

Recognizing simplicity of black-box groups and the frequency of p-singular elements in affine groups

Babai¹,

Shalev²

2001

Groups and Computation III

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Abstract. We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G?While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non-abelian matrix group remains open.For the purposes of polynomial time computation, we reduce this problem to the following "affine case:" Let A be an elementary abelian p-group which is a non-central minimal normal subgroup of a finite group G. Assume further that the quotient G/A is a finite simple group S of Lie type of characteristic p.The algorithmic goal is to distinguish the simple group S from the nonsimple group G. Both S and G are given as "black-box groups of characteristic p."The reduction to this basic problem involves a large number of recent techniques and results which we review along the way.We address the affine case for the groups S = PSL(2, q) where q = p f , p prime. We show that PSL(2, p) can be recognized in Monte Carlo polynomial time among all black-box groups of known characteristic. The situation for f ≥ 2 seems much harder; we exhibit challenging open cases for every f ≥ 2 when q is large.Along with S = PSL(2, p), the positive result holds for all simple groups S of Lie type of characteristic p with the property that in every nontrivial S-module in characteristic p, every element of S has a fixed point. We call these groups "unisingular." Very recently, Guralnick, Saxl, and Tiep classified these groups. They found that a large number of classes of simple groups are unisingular, including certain classes of linear and unitary groups, all symplectic groups over an odd prime field, all orthogonal groups of odd degree over an odd prime field, many orthogonal groups of even degree, and a number of classes of exceptional groups.2 László Babai and Aner Shalev

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“…Babai et al [8] An implementation developed by Malle and O'Brien is distributed with GAP and Magma. It includes naming procedures for the other quasisimple groups: if the non-abelian composition factor is alternating or sporadic, then we identify it by considering the orders of random elements.…”

Section: Black-box Groups Of Lie Typementioning

confidence: 99%