Black box classical groups

Kantor, William M.; Seress, Ákos

doi:10.1090/memo/0708

Cited by 61 publications

(168 citation statements)

References 12 publications

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“…Combining our machinery with Conder et al [23] and with [34] and the work by Brooksbank and Kantor [18,19,20,21], we obtain the following result. Theorem 2.9.…”

Section: Constructive Recognitionsupporting

confidence: 59%

Polynomial-time theory of matrix groups

Babai

Beals

Seress

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles.Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log.One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity).As a by-product, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N , does the group generated by A have a semisimple quotient of order ≥ N ?We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.

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“…Combining our machinery with Conder et al [23] and with [34] and the work by Brooksbank and Kantor [18,19,20,21], we obtain the following result. Theorem 2.9.…”

Section: Constructive Recognitionsupporting

confidence: 59%

Polynomial-time theory of matrix groups

Babai

Beals

Seress

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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“…One of the main ingredients comes from an unusual source: "black box group theory" [KS00], via a line of thought represented by [Br00, PW04, AlBo01, Bo02]. The subject as a whole deals with the problem of computing in large finite groups which are given in such a form that one can extract elements at random, and perform limited operations or tests on these elements.…”

Section: Afterword: Black-box Groupsmentioning

confidence: 99%

Involutions in groups of finite Morley rank of degenerate type

Borovik

Burdges

Cherlin

2007

Sel. math., New ser.

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“…Theorem 4.14 ( [KS1]). Let C p denote the class of classical finite simple groups of characteristic p. Then one can, in Monte-Carlo polynomial time, constructively recognize black-box members of C p within C p assuming the field of definition is tiny.…”

Section: Algorithm (C Leedham-green)mentioning

confidence: 99%

“…"Constructive recognition" means that an explicit isomorphism is constructed with a standard matrix representation of the group in question. As pointed out in [KS1], it follows, using Steinberg's presentations, that a presentation in terms of generators and relations can also be constructed in Monte Carlo polynomial time. This observation immediately upgrades the Monte Carlo algorithm of Theorem 4.14 to a Las Vegas algorithm, i. e., a Monte Carlo algorithm which does not err but is allowed to report failure with a small probability.…”

Section: Algorithm (C Leedham-green)mentioning

confidence: 99%

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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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Black box classical groups

Cited by 61 publications

References 12 publications

Polynomial-time theory of matrix groups

Polynomial-time theory of matrix groups

Involutions in groups of finite Morley rank of degenerate type

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

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