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A polynomialtime theory of blackbox groups I
, 1998
"... We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic o ..."
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Cited by 40 (6 self)
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We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomialtime randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “blackbox groups ” under minimal assumptions. In a blackbox group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”
A Recognition Algorithm for Classical Groups over Finite Fields
 Proc. London Math. Soc
, 1998
"... 2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126 ..."
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Cited by 29 (0 self)
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2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126
Simple Groups in Computational Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICANS
, 1998
"... This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation. ..."
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Cited by 12 (2 self)
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This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation.
Prime power graphs for groups of Lie type
 JOURNAL OF ALGEBRA
, 2002
"... We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. ..."
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Cited by 10 (6 self)
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We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. We prove that for any nite simple group G of Lie type, input as a black box group with an oracle to compute the orders of group elements, (G) and the characteristic of G can be computed by a Monte Carlo algorithm in time polynomial in the input length. The characteristic is needed as part of the input in a previous constructive recognition algorithm for G.
Deciding Finiteness for Matrix Groups over Function Fields
 Israel Journal of Mathematics
, 1998
"... Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n, F(t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n, F), for which the results of [1] ..."
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Cited by 8 (1 self)
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Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n, F(t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n, F), for which the results of [1] can be applied. When F is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more e#cient randomized algorithms. 1 Introduction Recently, computational group theory has directed increased attention to the development of algorithms for studying matrix groups. In particular, various recognition algorithms are of importance. The...
Permutation group algorithms via black box recognition algorithms
 in ‘‘Groups St. Andrews 1997 in Bath, vol. II,’’ London KANTOR AND SERESS �KS3� Math. Soc. Lect. Note Series 261
, 1999
"... If a black box simple group is known to be isomorphic to a classical group over a field of known characteristic, a Las Vegas algorithm is used to produce an explicit isomorphism. This is used to upgrade all nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the ..."
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Cited by 6 (5 self)
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If a black box simple group is known to be isomorphic to a classical group over a field of known characteristic, a Las Vegas algorithm is used to produce an explicit isomorphism. This is used to upgrade all nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to an exceptional group of Lie type or a 3dimensional unitary group. Key words and phrases: computational group theory, black box groups, classical groups, matrix group recognition
Constructive recognition of a black box group isomorphic to GL(n, 2
 In Groups and Computation II, DIMACS Series in Discrete Mathematics and Computer Science
, 1997
"... algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n,2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data str ..."
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Cited by 4 (0 self)
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algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n,2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data structure used to compute the isomorphism and each invocation of the isomorphism require polynomial time complexity. The ultimate goal is to eventually extend this result to the larger question of constructing the natural representation of classical groups. Extensions of the methods developed in this paper are discussed as well as open questions. 1.
Probabilistic group theory
 C. R. Math. Acad. Sci. Canada
, 2002
"... This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems i ..."
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This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems in group theory. It concludes with a brief summary of related results for infinite groups. Cet article donne un apercu sur trois aspects des façons dont la probabilité est appliquée à la théorie des groupes finis: les faits probabilistiques des groupes; la construction d’algorithmes aléatoires dans la computation; et l’application des moyens probabilistiques pour obtenir les theorems déterministiques dans la théorie des groupes. On termine avec un bref sommaire de resultats se rapportant aux groupes infinis.