Testing modules for irreducibility

Holt, Derek F.; Rees, Sarah

doi:10.1017/s1446788700036016

Cited by 102 publications

(134 citation statements)

References 4 publications

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“…It now suffices to explicitly map T * i to H and perform constructive recognition in the image. We compute a composition series of F d p , considered as a T * i module [44,30,32]. It follows from Proposition 8.2 that on one of the composition factors T * i acts as a quasisimple matrix group (cf.…”

Section: Arbitrary Characteristicmentioning

confidence: 99%

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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Section: Arbitrary Characteristicmentioning

confidence: 99%

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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“…(ii) For each maximal ideal M of R there is a unique p A W such that M ¼ ker p. We apply the MeatAxe [10], [12] to S to find the set X of irreducible S-submodules of V :¼ V =pV . As S is semisimple, X is a direct decomposition of V .…”

Section: 3mentioning

confidence: 99%

“…Hence we obtain a Wedderburn complement decomposition R ¼ S l JðRÞ. As S is semisimple its action on V is completely reducible and the MeatAxe [10], [12] finds a decomposition V ¼ V 1 l Á Á Á l V l as above. For each W A W, the map p W is a ring homomorphism as t is a ring homomorphism and W is an S-module.…”

Section: 3mentioning

confidence: 99%

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Finding central decompositions of p-groups

Wilson¹

2009

Journal of Group Theory

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Abstract. A Las Vegas polynomial-time algorithm is given to find a central decomposition of maximum size for a finite p-group of class 2. The proof introduces an associative Ã-ring as a tool for studying central products of p-groups. This technique leads to a translation of the problem into classical linear algebra which can be solved by application of the MeatAxe and other established module-theoretic algorithms. When p is small, our algorithm runs in deterministic polynomial time.

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“…The second is to deal with irreducible cases, which in this case are the finite simple groups. This paper addresses constructive recognition of matrix groups from 'both ends': on the one hand, we give an improved analysis of the Norton irreducibility test, part of the MEAT-AXE algorithm (see [1]), by providing a lower bound of the form a 1 − a 2 q −bc for the proportion of primary cyclic matrices in M(c, q b ), where a 1 , a 2 are constants depending only on q, b. To achieve this, we generalise the Kung-Stong cycle index (see [2,7]) to compute a generating function for the proportion.…”

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confidence: 99%

Estimation and Computation With Matrices Over Finite Fields

Corr¹

2014

Bull. Aust. Math. Soc.

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The matrix group recognition project is a worldwide effort to produce efficient algorithms for working with arbitrary matrix groups over finite fields (see [3,6]). Such groups are potentially very large in comparison to the input length, and dealing with them using deterministic methods is impractical.When a generating set for a group is input into a computer, a constructive recognition algorithm names the group and finds an efficient mapping between the input generators and a set of 'standard generators'. This allows various important questions to be answered quickly. Constructive recognition is a major natural goal in computational group theory.To recognise an arbitrary group, there are two tasks to perform. The first is to decompose the group into smaller components if possible, and work recursively. The second is to deal with irreducible cases, which in this case are the finite simple groups. This paper addresses constructive recognition of matrix groups from 'both ends': on the one hand, we give an improved analysis of the Norton irreducibility test, part of the MEAT-AXE algorithm (see [1]), by providing a lower bound of the form a 1 − a 2 q −bc for the proportion of primary cyclic matrices in M(c, q b ), where a 1 , a 2 are constants depending only on q, b. To achieve this, we generalise the Kung-Stong cycle index (see [2,7]) to compute a generating function for the proportion.On the other hand, we solve a particular family of base cases for the constructive recognition recursion, by extending the work of Magaard et al.[4] to provide a Las Vegas algorithm for constructive recognition of classical groups in irreducible representations of moderate degree. When the degree of the representation is large, existing black-box methods are effective. On the other hand, when the degree is equal to the natural degree, there are specific methods to address the problem.

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Testing modules for irreducibility

Cited by 102 publications

References 4 publications

Polynomial-time theory of matrix groups

Polynomial-time theory of matrix groups

Finding central decompositions of p-groups

Estimation and Computation With Matrices Over Finite Fields

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