Finite groups, 2-generation and the uniform domination number

Autor: Scott Harper, Timothy C. Burness
Rok vydání: 2020
Předmět:
Zdroj: Burness, T C & Harper, S 2020, ' Finite groups, 2-generation and the uniform domination number ', Israel Journal of Mathematics, vol. 239, pp. 271-367 . https://doi.org/10.1007/s11856-020-2050-8
University of St Andrews CRIS
ISSN: 1565-8511
0021-2172
DOI: 10.1007/s11856-020-2050-8
Popis: 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.
Comment: 60 pages; to appear in Israel Journal of Mathematics
Databáze: OpenAIRE
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