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• W2898286603 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 $G = langle x_i, yrangle$ 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." @default.
• W2898286603 created "2018-11-02" @default.
• W2898286603 creator A5024944956 @default.
• W2898286603 creator A5087244891 @default.
• W2898286603 date "2018-10-29" @default.
• W2898286603 modified "2023-09-25" @default.
• W2898286603 title "Finite groups, 2-generation and the uniform domination number" @default.
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• W2898286603 hasPublicationYear "2018" @default.
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