Finite groups, 2-generation and the uniform domination number

• Published in: Israel journal of mathematics 2020-08, Vol.239 (1), p.271-367
• Main Authors: Burness, Timothy C., Harper, Scott
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Conjugates; Group Theory and Generalizations; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Probabilistic methods; Statistical analysis; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-020-2050-8

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 ,…, x k , there exists y G ∈ G such that G = 〈 x i , y 〉 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 ) ⩾ 2 for every non-abelian finite simple group G . For any group with u ( G ) ⩾ 1, we define the uniform domination number γ u (G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈 x, y 〉 (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 γ 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 γ 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.