# On the Saxl graph of a permutation group

@article{Burness2017OnTS, title={On the Saxl graph of a permutation group}, author={Timothy C. Burness and Michael Giudici}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, year={2017}, volume={168}, pages={219 - 248} }

Abstract Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph Σ(G), which we call the Saxl graph of G. The vertices of Σ(G) are the points of Ω, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of Σ(G) for a finite transitive group G, as well as its diameter…

## 9 Citations

### On the Saxl graphs of primitive groups with soluble stabilisers

- MathematicsAlgebraic Combinatorics
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Let G be a transitive permutation group on a ﬁnite set Ω and recall that a base for G is a subset of Ω with trivial pointwise stabiliser. The base size of G , denoted b ( G ), is the minimal size of…

### On valency problems of Saxl graphs

- MathematicsJournal of Group Theory
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Abstract Let 𝐺 be a permutation group on a set Ω, and recall that a base for 𝐺 is a subset of Ω such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the…

### Saxl graphs of primitive affine groups with sporadic point stabilisers

- Mathematics
- 2021

Let G be a permutation group on a set Ω. A base for G is a subset of Ω whose pointwise stabiliser is trivial, and the base size of G is the minimal cardinality of a base. If G has base size 2, then…

### C O ] 1 0 A ug 2 02 0 On the Burness-Giudici Conjecture

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Let G be a permutation group on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. By b(G) we denote the size of the smallest base of G. Every permutation group with…

### On base sizes for primitive groups of product type

- MathematicsJournal of Pure and Applied Algebra
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### Base sizes for primitive groups with soluble stabilisers

- MathematicsAlgebra & Number Theory
- 2021

Let $G$ be a finite primitive permutation group on a set $\Omega$ with point stabiliser $H$. Recall that a subset of $\Omega$ is a base for $G$ if its pointwise stabiliser is trivial. We define the…

### On the Burness-Giudici Conjecture

- Mathematics
- 2020

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. By $b(G)$ we denote the size of the smallest base of $G$. Every…

### Finite groups, 2-generation and the uniform domination number

- MathematicsIsrael Journal of Mathematics
- 2020

Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G =…

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