Finite groups, 2-generation and the uniform domination number

Burness, Timothy C.; Harper, Scott

doi:10.1007/s11856-020-2050-8

Cited by 22 publications

(33 citation statements)

References 54 publications

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“…Since B n (G k , X) grows exponentially with n, the set {γ −1 m t k γ m : m ∈ N} can be considered as 'small' in G k . Note also the similarity to what is known, from [8], for finite simple groups. In that world, there is a 'small' set of elements S -in certain cases of size 2 -such that for every g ∈ G \ {1} there exists an s ∈ S such that g, s = G.…”

Section: Remark 45supporting

confidence: 66%

“…If there is a set X such that s X (G) 1, then X is called a total dominating set of G. We can then define the uniform spread of G as the supremum of {s C (G) : C a conjugacy class of G}, and denote this by u(G). The concept of uniform spread was introduced in [4]; both [7,8] provide further interesting results for finite groups. Note that u(G) s(G) by definition.…”

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: Remark 45supporting

confidence: 66%

Section: Introductionmentioning

confidence: 99%

On the spread of infinite groups

Cox¹

2022

Proceedings of the Edinburgh Mathematical Society

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“…Similarly, the case (n, q) = (3, 5) can be checked using Magma. In each of the remaining cases, the uniform domination of number of G is equal to 2 (see [10,Theorem 6(ii)]) and the result follows as in the proof of the previous proposition.…”

Section: It Remains To Verify the Bound δ S (G)mentioning

confidence: 74%

On the soluble graph of a finite group

Burness¹,

Lucchini²,

Nemmi³

2021

Preprint

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Let G be a finite insoluble group with soluble radical R(G). In this paper we investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G \ R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS (G), is at most 5. More precisely, we show that δS (G) 3 if G is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS (Sn) = 3 for all n 6. We also establish the existence of simple groups with δS (G) 4. For instance, we prove that δS (A2p+1) 4 for every Sophie Germain prime p 5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.

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“…Fix a divisor ℓ of n with 1 < ℓ < n and identify Ω with Π ℓ , the set of partitions of [n] into ℓ parts of equal size. Note that H = (S n/ℓ ≀ S ℓ ) ∩ G. If 3 r ℓ then [19,Lemma 4.5] implies that fpr(x) < ℓ −2 and the result follows. Similarly, if r = 2 then the bound in [19,Lemma 4.6] is sufficient unless x is a transposition and ℓ = 2, which is a genuine exception, as noted above.…”

Section: Almost Simple Groups With Non-classical Soclementioning

confidence: 86%

Fixed point ratios for finite primitive groups and applications

Burness¹,

Guralnick²

2021

Preprint

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Let G be a finite primitive permutation group on a set Ω and recall that the fixed point ratio of an element x ∈ G, denoted fpr(x), is the proportion of points in Ω fixed by x. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing fpr(x) with the order of x. Our main theorem classifies the triples (G, Ω, x) as above with the property that x has prime order r and fpr(x) > 1/(r + 1). There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree m with minimal degree at most 2m/3. Secondly, our main result plays a key role in recent work of the authors (together with Moretó and Navarro) on the commuting probability of p-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava.

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Finite groups, 2-generation and the uniform domination number

Cited by 22 publications

References 54 publications

On the spread of infinite groups

On the spread of infinite groups

On the soluble graph of a finite group

Fixed point ratios for finite primitive groups and applications

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