Infinite 32‐generated groups

Donoven, Casey; Harper, Scott

doi:10.1112/blms.12356

Cited by 5 publications

(5 citation statements)

References 25 publications

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“…I thank Scott Harper of the University of Bristol for his illuminating talks, ever-keen interest to discuss group theory, and comments on this work. I also thank both him and Casey Donoven for the interesting questions they raised in [13]. I thank the anonymous referees for their comments.…”

Section: Conjugate Gmentioning

confidence: 91%

“…Note that there exist finitely generated, but not 2-generated, infinite simple groups, as shown in [14]. This leads us to the first question raised in [13].…”

Section: Introductionmentioning

confidence: 99%

“…[2-4, 8, 9, 16, 17]. But the first paper on spread for infinite groups, [13], only recently appeared. They showed that the Higman-Thompson and Brin-Thompson groups are 3 2 -generated, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…We then investigate the other questions of [13], which require additional notions. Given a group G, let us say that the spread on X ⊆ G is the supremum of those k ∈ N ∪ {0} such that for any g 1 , .…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Conjugate Gmentioning

confidence: 91%

“…Note that there exist finitely generated, but not 2-generated, infinite simple groups, as shown in [14]. This leads us to the first question raised in [13].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the spread of infinite groups

Cox¹

2022

Proceedings of the Edinburgh Mathematical Society

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“…In fact, there even exist finitely generated simple groups that are not 2-generated (see [34]). However, recent work of Donoven and Harper [29] shows that Thompson's group V , and related infinite families of finitely presented groups, are It is natural to ask if the cyclic quotient property is equivalent to 3 2 -generation for finite groups. This is a conjecture of Breuer, Guralnick and Kantor (see [10,Conjecture 1.8]).…”

Section: Introductionmentioning

confidence: 99%

The spread of a finite group

Burness,

Guralnick,

Harper

2020

Preprint

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A group G is said to be 3 2 -generated if every nontrivial element belongs to a generating pair. It is easy to see that if G has this property then every proper quotient of G is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if G is a finite group and every proper quotient of G is cyclic, then for any pair of nontrivial elements x1, x2 ∈ G, there exists y ∈ G such that G = x1, y = x2, y . In other words, s(G) 2, where s(G) is the spread of G. Moreover, if u(G) denotes the more restrictive uniform spread of G, then we can completely characterise the finite groups G with u(G) = 0 and u(G) = 1. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.

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Introduction

Harper

2021

Lecture Notes in Mathematics

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Infinite 32‐generated groups

Cited by 5 publications

References 25 publications

On the spread of infinite groups

On the spread of infinite groups

The spread of a finite group

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