Asymptotic Results for Primitive Permutation Groups

Pyber, László; Shalev, Aner

doi:10.1006/jabr.1996.6818

Cited by 17 publications

(19 citation statements)

References 36 publications

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“…It can be shown that this number is bounded above by n c log n for all n, and bounded below by n c$ log nÂlog log n for infinitely many values of n [PSh2]. See also [P], [PSh1] for related results, and for applications of estimates of this type to subgroup growth of infinite groups.…”

Section: Introductionmentioning

confidence: 96%

Maximal Subgroups of Symmetric Groups

Liebeck

Shalev

1996

Journal of Combinatorial Theory, Series A

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We show that S n has at most n 6Â11+o(1) conjugacy classes of primitive maximal subgroups. This improves an n c log 3 n bound given by Babai. We conclude that the number of conjugacy classes of maximal subgroups of S n is of the form ( 1 2 +o(1))n. It also follows that, for large n, S n has less than n ! maximal subgroups. This confirms a special case of a conjecture of Wall. Improving a recent result from [MSh], we also show that any finite almost simple group has at most n 17Â11+o(1) maximal subgroups of index n.

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Section: Introductionmentioning

confidence: 96%

Maximal Subgroups of Symmetric Groups

Liebeck

Shalev

1996

Journal of Combinatorial Theory, Series A

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“…By [34], if G is a primitive permutation group of degree n > 2, then there is a constant a such that d(G) ≤ a log n/ √ log log n. Now, applying Theorem 8.1, we obtain the desired result. Corollary 8.2 improves an n c log n bound which is the main result of [49]. Note that for infinitely many positive integers n even, the number of isomorphism types of primitive soluble groups is at least n log n log log n [49].…”

Section: Corollary 82mentioning

confidence: 71%

“…Corollary 8.2 improves an n c log n bound which is the main result of [49]. Note that for infinitely many positive integers n even, the number of isomorphism types of primitive soluble groups is at least n log n log log n [49].…”

Section: Corollary 82mentioning

confidence: 71%

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Random generation of finite and profinite groups and group enumeration

Jaikin–Zapirain

Pyber

2011

Ann. Math.

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We obtain a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice.Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal.As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most n cr as conjectured by Mann.

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“…We refer to the surveys of Cameron [4] and Pyber [17,18] and the recent paper by Pyber and Shalev [19] for a detailed exposition of this subject. In this paper we concentrate our attention on the number of generators.…”

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confidence: 99%

Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups

2000

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A well-developed branch of asymptotic group theory studies the properties of classes of linear and permutation groups as functions of their degree. We refer to the surveys of Cameron [4] and Pyber [17,18] and the recent paper by Pyber and Shalev [19] for a detailed exposition of this subject. In this paper we concentrate our attention on the number of generators. Our results, like most recent results in this area, depend on the classification of finite simple groups (which will be referred to hereafter as CFSG).Concerning linear groups, in 1991, Kovács and Robinson [11] proved that every finite completely reducible linear group of dimension d can be generated by and afterwards, in 1993, Bryant, et al. [3] proved the following result: to each field F whose degree over its prime subfield is finite,

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Asymptotic Results for Primitive Permutation Groups

Abstract: We prove that the number of conjugacy classes of primitive permutation groups cŽ n.

Cited by 17 publications

References 36 publications

Maximal Subgroups of Symmetric Groups

Maximal Subgroups of Symmetric Groups

Random generation of finite and profinite groups and group enumeration

Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups

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