Random generation of finite and profinite groups and group enumeration

Jaikin–Zapirain, Andrei; Pyber, László

doi:10.4007/annals.2011.173.2.4

Cited by 32 publications

(64 citation statements)

References 48 publications

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“…Remarkable results characterizing PFG profinite groups have been very recently obtained by Jaikin-Zapirain and Pyber [26]. Theorem 1 in that paper provides strong bounds on ν(G) for G finite.…”

Section: By [42 12] We Have M(g) < ν(G) + 4 For Any Finite Group Gmentioning

confidence: 76%

Generation and random generation: From simple groups to maximal subgroups

Burness¹,

Liebeck

Shalev

2013

Advances in Mathematics

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Abstract. Let G be a finite group and let d(G) be the minimal number of generators for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We prove that d(H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.

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Section: By [42 12] We Have M(g) < ν(G) + 4 For Any Finite Group Gmentioning

confidence: 76%

Generation and random generation: From simple groups to maximal subgroups

Burness¹,

Liebeck

Shalev

2013

Advances in Mathematics

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“…We find a useful way of characterising profinite groups with this property. This characterisation allows us to actually work with the APFG property and it is not difficult to prove, by using the different characterisations of PFG groups given in [17], [16] and [10], that PFG groups are APFG. It is worth mentioning that the converse does not hold, as is shown in Example 4.5; moreover, we produce an example of a finitely generated profinite group which is not APFG.…”

Section: Theorem 2: Let G Be a Countably-based Profinite Group And Asmentioning

confidence: 97%

The generation of the augmentation ideal in profinite groups

Damian

2011

Isr. J. Math.

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We find a formula for computing the minimum number of generators for the augmentation ideal in the profinite setting; this is a generalisation of a result obtained in the finite case by Cossey, Gruenberg and Kovács.We then consider probabilistic questions connected with the generation of the augmentation ideal and we define the class of APFG groups as those profinite groups for which the probability of generating the augmentation ideal with t random elements is non-zero for some t ∈ N. We give a characterisation of these groups which allows us easily to compare APFG groups with PFG groups and we show that PFG groups are APFG but we give examples showing that this is a strict inclusion.

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“…As previously noted, δ(H) < 1 for all H ∈ M 1 (G), so The constant γ here can be expressed in terms of the undetermined constant β in Theorem 4.10. It would be desirable to have an effective result with an explicit constant, but it seems rather difficult to extract constants from the proof of Theorem 4.10 in [33].…”

Section: Subgroup Growthmentioning

confidence: 99%

Simple Groups, Generation and Probabilistic Methods

Burness¹

2019

Groups St Andrews 2017 in Birmingham

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It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, (a, b)-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

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Random generation of finite and profinite groups and group enumeration

Cited by 32 publications

References 48 publications

Generation and random generation: From simple groups to maximal subgroups

Generation and random generation: From simple groups to maximal subgroups

The generation of the augmentation ideal in profinite groups

Simple Groups, Generation and Probabilistic Methods

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