Computing the composition factors of a permutation group in polynomial time

Luks, Eugene M.

doi:10.1007/bf02579204

Cited by 42 publications

(19 citation statements)

References 13 publications

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“…Moreover, the same algorithms also construct a presentation of the group. We shall also need the fact that for permutation groups, a composition chain can be found in polynomial time [39].…”

Section: General Theorymentioning

confidence: 99%

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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: General Theorymentioning

confidence: 99%

“…The first analyzed version [27], motivated by the first group theoretic algorithm in graph isomorphism testing [3], appeared in 1980. Subsequently a number of advanced questions were also solved in polynomial time, including composition factors [39], Sylow subgroups [33] (cf. [41]), and the solvable radical ( [38], see [42]).…”

Section: Introductionmentioning

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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“…(Initially, we know a normal series of G/G.) Using the methods for the computation of composition series [16,14,1], we obtain a subgroup U H such that T := H/U is simple. Let…”

Section: Computing a Chief Seriesmentioning

confidence: 99%

Conjugacy classes in finite permutation groups via homomorphic images

Hulpke

1999

Math. Comp.

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“…For the more complicated task of computing a composition series, Theorem 1.3 gives an improvement of five orders of magnitude from Luks's original algorithm [Lu87]. This result requires a deeper probe into the structure of primitive groups with different types of socle, in the spirit of the O'NanScott theorem [Sc], [Cam].…”

Section: (B) Finding a Composition Series Of Gmentioning

confidence: 99%

Fast Management of Permutation Groups I

Babai¹,

Luks²,

Seress³

1997

SIAM J. Comput.

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Abstract. We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.Key words. permutation group algorithm, strong generating set AMS subject classifications. 68Q40, 20B40PII. S00975397942294171. Introduction. Since the size of a permutation group G on n letters can be exponential in n, it is customary, for computational purposes, to specify G by a small list of generators. However, the succinctness of such a representation raises the issue of whether we can deal effectively with the groups that we can specify. Can one, for example, find the order of G and test membership in G without enumerating all of its elements?In fact, in the late sixties, Sims developed efficient algorithms for permutation group manipulation [Si70]. These included the key notion of a strong generating set (SGS) which is the underlying concept in essentially all polynomial-time algorithms in computational group theory. Given a chain G = G 0 ≥ G 1 ≥ · · · ≥ G m = 1 of subgroups of G, an SGS with respect to this chain is a set T ⊂ G such that T ∩ G i generates G i for each i. Sims's algorithm uses the point stabilizer chain; that is, G i is the pointwise stabilizer of the first i points of the permutation domain. While Sims's methods for constructing an SGS have been widely used in computational group theory since their inception, the question of their asymptotic efficiency was not resolved until 1980. Furst, Hopcroft, and Luks [FHL] observed that a version of Sims's algorithm runs in polynomial time, namely O(n 6 + sn 2 ) steps, where s is the number of generators given for G. Subsequently, Knuth [Kn] and Jerrum [Je82], [Je86] gave variants with running time O(n 5 + sn 2 ). All of these algorithms rest on the most elementary group theory.

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Computing the composition factors of a permutation group in polynomial time

Abstract: Giveng enerators for a group of permutations, it is shown that generators for the subgroups in a composition series can be found in polynomial time. The procedure also yields permutation representations of the composition factors.

Cited by 42 publications

References 13 publications

Polynomial-time theory of matrix groups

Polynomial-time theory of matrix groups

Conjugacy classes in finite permutation groups via homomorphic images

Fast Management of Permutation Groups I

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