Finite groups, 2-generation and the uniform domination number

TL;DR

This paper investigates the properties of finite groups related to their generation by conjugate elements, focusing on the uniform domination number, and makes progress on classifying simple groups with minimal uniform dominating sets.

Contribution

It introduces new bounds and classifications for the uniform domination number and spread of finite groups, especially simple groups, and explores their probabilistic properties.

Findings

Progress towards classifying simple groups with uniform domination number 2

New bounds on the probability that two conjugates form a uniform dominating set

Results on 2-generation of soluble and symmetric groups

Abstract

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 $G = \langle x_i, y\rangle$ for all $i$. The more restrictive notion of uniform spread, denoted $u(G)$, requires $y$ to be chosen from a fixed conjugacy class of $G$, and a theorem of Breuer, Guralnick and Kantor states that $u(G) \geqslant 2$ for every non-abelian finite simple group $G$. For any group with $u(G) \geqslant 1$, we define the uniform domination number $\gamma_u(G)$ of $G$ to be the minimal size of a subset $S$ of conjugate elements such that for each nontrivial $x \in G$ there exists $y \in S$ with $G = \langle x, y \rangle$ (in this situation, we say that $S$ is a uniform dominating set for $G$). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on $\gamma_u(G)$ for all simple groups $G$. In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups $G$ with $\gamma_u(G)=2$, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for $G$. We also establish new results concerning the $2$-generation of soluble and symmetric groups, and we present several open problems.