On cubic graphical regular representations of finite simple groups

Xia, Binzhou

doi:10.1016/j.jctb.2019.06.002

Cited by 10 publications

(5 citation statements)

References 19 publications

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“…It is known by [6] that the non-simple group PSU 3 (2) does not admit cubic GRRs, and it is clear that PSU 3 (3) has no cubic GRR since the group cannot be generated by any triple of involutions. Inspired by the work on cubic GRRs of PSL 3 (q) in [18], in this paper, we construct cubic GRRs of PSU 3 (q) for each q ≥ 4 and verify Conjecture 1.1 for PSU 3 (q), which eventually helps to confirm Conjecture 1.1 in [20].…”

Section: Introductionmentioning

confidence: 86%

“…By [3,Theorem 4,Corollary 6], for G = PSL 3 (q) or PSU 3 (q) and any pair of generators (x, y) of G where y is an involution, Aut(G, {x, x −1 , y}) is always nontrivial and therefore the connection set of any cubic GRR of G (if it exists) consists of three involutions. In [20,Theorem 1.5], Xia, Zheng and Zhou showed that Spiga's conjecture ( [16,Conjecture 1.3]) can be saved by adding PSL 3 (q) and PSU 3 (q) to the list of exceptional groups.…”

Section: Introductionmentioning

confidence: 99%

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Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

Li¹,

Xia²,

Zhang³

et al. 2022

Preprint

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A graphical regular representation (GRR) of a group G is a Cayley graph of G whose full automorphism group is equal to the right regular permutation representation of G. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that PSU 3 (q) has a cubic GRR if and only if q ≥ 4. Moreover, a cubic GRR of PSU 3 (q) is constructed for each of these q.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

Li¹,

Xia²,

Zhang³

et al. 2022

Preprint

Self Cite

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“…Given a finite group X, it is natural to consider the existence of a DRR with a prescribed valency, noting that the valency of Cay(𝑋, 𝑆) is |𝑆|. Recently, there are some results concerning this problem in relation to finite simple groups (for example, see [58,61] for the existence of some families of DRRs with a fixed valency 𝑘 3, and [60] for 𝑘 5). However, there appear to be no asymptotic results in the literature concerning the proportion of DRRs of a fixed valency of a given finite group.…”

Section: Introductionmentioning

confidence: 99%

Base sizes of primitive groups of diagonal type

Huang

2023

Forum of Mathematics, Sigma

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Let G be a permutation group on a finite set $\Omega $ . The base size of G is the minimal size of a subset of $\Omega $ with trivial pointwise stabiliser in G. In this paper, we extend earlier work of Fawcett by determining the precise base size of every finite primitive permutation group of diagonal type. In particular, this is the first family of primitive groups arising in the O’Nan–Scott theorem for which the exact base size has been computed in all cases. Our methods also allow us to determine all the primitive groups of diagonal type with a unique regular suborbit.

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“…However, at this stage, a Sabidussi-like theorem concerning GRRs of a prescribed valency is far out of reach. Even in the special case that the valency is 3, although it has attracted attention from several authors over the last few decades [2,4,9,12,16,19,20,21,22], little is yet known of which groups have a cubic GRR.…”

Section: Introductionmentioning

confidence: 99%