On the uniform spread of almost simple symplectic and orthogonal groups

Harper, Scott

doi:10.1016/j.jalgebra.2017.07.008

Cited by 21 publications

(30 citation statements)

References 37 publications

(107 reference statements)

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“…A finite group G is 3 2 -generated if and only if every proper quotient of G is cyclic. For recent work towards proving this conjecture, see [12,19].…”

Section: Conjecturementioning

confidence: 99%

Infinite $\frac{3}{2}$-generated groups

Donoven,

Harper

2019

Preprint

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Every finite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be 3 2 -generated. Thompson's group V was the first finitely presented infinite simple group to be discovered. The Higman-Thompson groups V n and the Brin-Thompson groups mV are two families of finitely presented groups that generalise V. In this paper, we prove that all of the groups V n , V ′ n and mV are 3 2 -generated. As far as the authors are aware, the only previously known examples of infinite noncyclic 3 2 -generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.

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“…A finite group G is 3 2 -generated if and only if every proper quotient of G is cyclic. For recent work towards proving this conjecture, see [12,19].…”

Section: Conjecturementioning

confidence: 99%

Infinite $\frac{3}{2}$-generated groups

Donoven,

Harper

2019

Preprint

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“…• G 0 ∈ {PSp n (q), Ω n (q) (nq odd)}: s(G) 2 by work of Scott Harper (2016) in his PhD thesis (see [58]). The goal is to use Corollary 3.11 to show that u(G) 2.…”

Section: Moreover the Interplay Between Groups And Graphs Suggests Mmentioning

confidence: 99%

Simple groups, fixed point ratios and applications

Burness

2018

Local Representation Theory and Simple Groups

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The study of fixed point ratios is a classical topic in permutation group theory, with a long history stretching back to the origins of the subject in the 19th century. Fixed point ratios arise naturally in many different contexts, finding a wide range of applications. In this survey article we focus on fixed point ratios for simple groups of Lie type, highlighting some of the main results, applications and related problems.

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“…In the case of finite groups, there has been an exceptional effort [6,9,18,19] to prove the converse of this statement, which was first conjectured in [4] and eventually completed in [9]. For infinite groups, the converse to the above lemma is false for a trivial reason: the existence of infinite simple groups that are not 2-generated.…”

Section: Introductionmentioning

confidence: 99%

On the spread of infinite groups

Cox¹

2022

Proceedings of the Edinburgh Mathematical Society

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A group is $\frac 32$ -generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$ -generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$ -generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$ . We then show that $G_1$ and $G_2$ are $\frac 32$ -generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$ , we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$ -generated whereas $G_{2k}$ is not.

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On the uniform spread of almost simple symplectic and orthogonal groups

Cited by 21 publications

References 37 publications

Infinite $\frac{3}{2}$-generated groups

Infinite $\frac{3}{2}$-generated groups

Simple groups, fixed point ratios and applications

On the spread of infinite groups

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