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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 47 (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.”
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
Some topics in asymptotic group theory
 Groups, Combinatorics and Geometry, volume 165 of LMS Lecture Notes
, 1992
"... There is nothing unusual about asymptotics in fmite group theory: there are a number of known (or even wellknown) asymptotic results. While these are not really the subject of this paper, it seems appropriate to begin with some especially intriguing examples (the fIrst and last of which will be nee ..."
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Cited by 6 (3 self)
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There is nothing unusual about asymptotics in fmite group theory: there are a number of known (or even wellknown) asymptotic results. While these are not really the subject of this paper, it seems appropriate to begin with some especially intriguing examples (the fIrst and last of which will be need later). 1.1. If P is prime then the number of isomorphism classes of groups of order pk.l. k3 _ 6k.l.f3 + 0(k8/3) is at least pZ7 (Higman [HiD, and asymptotically::: p2 (Sims [SiD.
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Cited by 1 (1 self)
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
Small Base Groups, Large Base Groups and the Case of Giants
"... Abstract. Let G be a finite permutation group acting on a set Γ. A base for G is a finite sequence of elements of Γ whose pointwise stabiliser in G is trivial. Most (families of) finite permutation groups admit a base that grows very slowly as the degree of the group increases. Such groups are known ..."
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Abstract. Let G be a finite permutation group acting on a set Γ. A base for G is a finite sequence of elements of Γ whose pointwise stabiliser in G is trivial. Most (families of) finite permutation groups admit a base that grows very slowly as the degree of the group increases. Such groups are known as small base and very efficient algorithms exist for dealing with them. However, some families of permutation groups, such as the symmetric groups, do not admit a small base. Dealing with these socalled large base groups is a fascinating area of current research. This thesis explores two closely interrelated strands of modern group theory. Initially, the focus is on identifying the large base primitive permutation groups, which can be achieved by making use of two landmark results in finite group theory: The Classification of Finite Simple Groups and the O’NanScott Theorem for primitive permutation groups. Focus then shifts to algorithmic aspects of large base groups, in particular to the family known as the giants. We cover details such as recognition of large base Galois groups, generation of random elements of finite groups and give details of the very new paradigm of algorithms for black box groups. We conclude with an investigation into the constructive recognition problem for large base black box groups.
COMPUTABLE SUBGROUP CHAINS AND SHADOWING
"... Abstract. We present a new structural framework for computational group theory, based on chains of subgroups. This extends existing methods, such as SchreierSims techniques for permutation groups. This framework is now a part of the GAP 4 computational algebra system. It will be useful for implemen ..."
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Abstract. We present a new structural framework for computational group theory, based on chains of subgroups. This extends existing methods, such as SchreierSims techniques for permutation groups. This framework is now a part of the GAP 4 computational algebra system. It will be useful for implementing the matrix group recognition project. Contents