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… 

On the Saxl graphs of primitive groups with soluble stabilisers

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

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

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

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

Base sizes for primitive groups with soluble stabilisers

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

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

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 =

References

SHOWING 1-10 OF 47 REFERENCES

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set Ω. A base for G is a subset B ⊆ Ω with pointwise stabilizer in G that is trivial; we write b(G) for the smallest size of a base for G. In this paper we

The base size of a primitive diagonal group

Partition Actions of Symmetric Groups and Regular Bipartite Graphs

A base of an action of a group G on a set Ω is a subset B ⊆ Ω such that the pointwise stabiliser of B in G is the identity. We prove that if Ω is the set of partitions of [1, kl] into l subsets of

Bases of primitive permutation groups

A base B for a finite permutation group G acting on a set Ω is a subset of Ω with the property that only the identity of G can fix every element of B. In this dissertation, we investigate some

On base sizes for symmetric groups

A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer of B in G is trivial. The base size of G, denoted by b(G), is the minimal cardinality of a base. Let G

On base sizes for algebraic groups

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this

Base sizes for S-actions of finite classical groups

Let G be a permutation group on a set Ω. A subset B of Ω is a base for G if the pointwise stabilizer of B in G is trivial; the base size of G is the minimal cardinality of a base for G, denoted by

Base sizes for sporadic simple groups

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise

On base sizes for actions of finite classical groups

Let G be a finite almost simple classical group and let Ω be a faithful primitive non‐standard G‐set. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. Let b(G) be the

On the Order of Uniprimitive Permutation Groups

One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for the