Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups

Lucchini, Andrea; Menegazzo, Federico; Morigi, Marta

doi:10.1006/jabr.1999.8081

Cited by 17 publications

(11 citation statements)

References 18 publications

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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].…”

Section: Corollary 82mentioning

confidence: 85%

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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Section: Corollary 82mentioning

confidence: 85%

Random generation of finite and profinite groups and group enumeration

Jaikin–Zapirain

Pyber

2011

Ann. Math.

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“…McIver and Neumann [81] showed that every subgroup of S n can be generated by ⌊n/2⌋ elements if n = 3, and by 2 if n = 3. This bound is best possible for arbitrary subgroups, but for transitive or primitive subgroups it has been improved in [71,72] to the statements in the theorem. Now we are going to prove that the minimum number of generators of any 2-homogeneous finite group is 2.…”

Section: Group Theorymentioning

confidence: 99%

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Araújo¹,

Bentz

Cameron

2018

Trans. Amer. Math. Soc.

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Let X be a finite set such that |X| = n, and let k < n/2. A group is k-homogeneous if it has only one orbit on the sets of size k. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate k-homogeneous groups (for k ≥ 1); in particular we show that 2-homogeneous groups are 2-generated. We also describe the orbits of k-homogenous groups on partitions with n − k parts, classify the 3-homogeneous groups G whose orbits on (n − 3)-partitions are invariant under the normalizer of G in S n , and describe the normalizers of 2-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

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“…Theorem 6.5. ( [27]) Let G ≤ S n be a primitive permutation group. Then the smallest number of elements needed to generate G is at most C log n √ log log n ,…”

Section: Problemsmentioning

confidence: 99%

Imprimitive permutations in primitive groups

et al. 2017

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The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. Lisboa,

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Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups

Cited by 17 publications

References 18 publications

Random generation of finite and profinite groups and group enumeration

Random generation of finite and profinite groups and group enumeration

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Imprimitive permutations in primitive groups

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