On the uniform domination number of a finite simple group

TL;DR

This paper introduces the uniform domination number of finite simple groups, providing bounds and demonstrating that infinitely many such groups have a uniform domination number of 2, using probabilistic and permutation group techniques.

Contribution

The paper defines the invariant mma_u(G), establishes bounds for it across simple groups, and connects it to permutation group bases, advancing understanding of generating sets.

Findings

mma_u(G)  2 for infinitely many simple groups

Probabilistic methods and fixed point ratios are effective in bounding mma_u(G)

Connections to permutation group bases enable application of recent base size results

Abstract

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $\gamma_u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s \rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $\gamma_u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $\gamma_u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups $G$ with $\gamma_u(G) = 2$. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.