On the uniform domination number of a finite simple group

Burness, Timothy C.; Harper, Scott

doi:10.1090/tran/7593

Cited by 16 publications

(50 citation statements)

References 44 publications

(137 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In each of the remaining cases, we can find an element in G of order r that is contained in a unique maximal subgroup H, where r and H are given below. To see this, we refer the reader to [9, Table 1], noting the correction µ(G) = 1 for G = Th since the same computational approach implemented in the proof of [9,Theorem 4.1] shows that elements of order 39 are indeed contained in a unique maximal subgroup. Then by inspecting [11] we see that b(G, H) = 2 in each case and this allows us to conclude via Lemma 2.6.…”

Section: Sporadic Groupsmentioning

confidence: 99%

“…Proof. Following [9], let γ u (G) be the uniform domination number of G. This is defined to be the minimal size of a set of conjugate elements {x 1 , . .…”

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

confidence: 99%

“…First assume G = Co 1 and fix x ∈ G of order 26. Then H = (A 4 × G 2 (4)):2 is the unique maximal subgroup of G containing x (see[9, Table 1]) and thus B 1 (x) is contained in H. With the aid of Magma, this observation allows us to construct B 1 (x) by working inside H (we get |B 1 (x)| = 1871) and then by random search we can find an element g ∈ G such that B 1 (x) ∩ B 1 (x g ) is empty. This implies that δ(x, x g ) 3 and the result follows.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On the soluble graph of a finite group

Burness¹,

Lucchini²,

Nemmi³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Sporadic Groupsmentioning

confidence: 99%

“…Proof. Following [9], let γ u (G) be the uniform domination number of G. This is defined to be the minimal size of a set of conjugate elements {x 1 , . .…”

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

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

On the soluble graph of a finite group

Burness¹,

Lucchini²,

Nemmi³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is therefore natural to ask the following. Following [14], a subset S ⊆ G is a total dominating set for G if for all x ∈ G there exists y ∈ S such that x, y = G. Therefore, u(G) > 0 if and only if G has a total dominating set consisting of conjugate elements, a so-called uniform dominating set.…”

Section: Questionsmentioning

confidence: 99%

Infinite 32‐generated groups

Donoven

Harper

2020

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

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 Vn 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 Vn, 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.

show abstract

“…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%

On the spread of infinite groups

Cox¹

2022

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

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.

show abstract

On the uniform domination number of a finite simple group

Cited by 16 publications

References 44 publications

On the soluble graph of a finite group

On the soluble graph of a finite group

Infinite 32‐generated groups

On the spread of infinite groups

Contact Info

Product

Resources

About