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

Babai, László; Shalev, Aner

doi:10.1515/9783110872743.39

Cited by 14 publications

(19 citation statements)

References 17 publications

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“…In Section 8 we discuss how to recognise a linear group isomorphic to SL(2, q) (as opposed to a projective linear group isomorphic to PSL(2, q)), and how to prove that the input group is in fact isomorphic, as projective linear group, to PSL(2, q). We also discuss a problem of Babai & Shalev [2]. Finally, we report on the performance of an implementation in Magma [8].…”

Section: Theorem 11 Let G Be a Subgroup Of Gl(d F ) For D ≥ 2 Whementioning

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: Theorem 11 Let G Be a Subgroup Of Gl(d F ) For D ≥ 2 Whementioning

confidence: 99%

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Conder

Leedham‐Green

O’Brien

2005

Trans. Amer. Math. Soc.

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“…This result allows p-regular elements to be discovered in S efficiently by random sampling, a key ingredient to several recent algorithms for matrix groups [2,4,3]. We emphasize that p may or may not be equal to the characteristic of the field of definition of S when S is of Lie type.…”

Section: Corollary 13 There Exists a Constant C > 0 Such That For Amentioning

confidence: 97%

“…This paper has been a long time in coming; versions of its main results have been quoted and applied in several papers, starting with [2] in 1999. Theorem 1.1 has been quoted as 'Theorem 4.9' in the paper [4] in the following form.…”

Section: Corollary 13 There Exists a Constant C > 0 Such That For Amentioning

confidence: 99%

On the Number ofp-Regular Elements in Finite Simple Groups

Babai¹,

Pálfy²,

Saxl³

2009

LMS J. Comput. Math.

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A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n − 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group A n , this proportion is at least 26/(27 √ n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary field F , if the simple group S is a quotient of a finite subgroup of GL n (F ) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998). Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for all n 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in P SL(n, q) is less than 3/ √ n.

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“…It summarises the work of [8] based on [26] and [2,13]. The answer is not known even if O p (X) is known to be a minimal normal subgroup in X and X/O p (X) a simple group of Lie type in characteristic p. Moreover, as shown in [10], the general problem can be reduced to this minimal configuration.…”

Section: Black Box Groups Of Odd Characteristicmentioning

confidence: 99%

Centralisers of involutions in black box groups

Borovik¹

2002

Computational and Statistical Group Theory

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We discuss basic structural properties of finite black box groups. A special emphasis is made on the use of centralisers of involutions in probabilistic recognition of black box groups. In particular, we suggest an algorithm for finding the p-core of a black box group of odd characteristic. This special role of involutions suggest that the theory of black box groups reproduces, at a non-deterministic level, some important features of the classification of finite simple groups.2000 Mathematical Subject Classification: 20P05. What is a black box group?A black box group X is a device or an algorithm ('oracle' or 'black box' ) which produces (nearly) uniformly distributed independent random elements from some finite group X. These elements are encoded as 0-1 strings of uniform length; given strings representing x, y ∈ X, the black box can compute strings representing xy and x −1 , and decide whether x = y in time bounded from above by a constant. In this setting, one is usually interested in finding probabilistic algorithms which allow us to determine, with probability of error ǫ, the isomorphism type of X in time O(|ǫ| · (log |X|) c ). We say in this situation that our algorithm is run in Monte Carlo polynomial time. A critical discussion of this concept can be found in [6], while [7] contains a detailed survey of the subject. See also the forthcoming book by Seress [39].In this paper we discuss a (still rather rudimentary) structural approach to the theory of black box group. We briefly survey methods for constructing black box oracles for subgroups and factor groups of black box groups, and then show how one can construct black box oracles for centralisers of involutions. They are used in the algorithm for finding the p-core of a black box group of characteristic p.Isomorphisms and homomorphisms of black box groups are understood as isomorphisms and homomorphisms of their underlying groups. However we reserve the term black box subgroup for a subgroup of a black box group endowed with its own black box oracle.Despite this rather abstract general setting, practically important black box groups usually appear as big permutation or matrix groups. For example, given two square matrices x and y of size, say, 100 by 100 over a finite field, it is unrealistic to list all elements in the group X generated by x and y and determine the isomorphism class of X by inspection. But this can often be done, with an arbitrarily small probability of error, by studying a sample of random products of the generators x and y. The explosive growth of the theory of black box groups in recent years is reflected in numerous publications (see, for example, the survey paper [29] on the computational matrix group project) and the fundamental work [26]), and algorithms implemented in the software packages GAP [22] and MAGMA [15]. Our observation (Section 2) that centralisers of involutions allow to compute unipotent radicals of black box groups of odd characteristic might be used in the computational matrix group project. This paper is...

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

Cited by 14 publications

References 17 publications

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

Constructive recognition of 𝑃𝑆𝐿(2,𝑞)

On the Number ofp-Regular Elements in Finite Simple Groups

Centralisers of involutions in black box groups

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