Finite groups, 2-generation and the uniform domination number (Q2204406)

Let \(G\) be a finite non-cyclic group that can be generated by two elements. In [Mich. Math. J. 22, 53--64 (1975; Zbl 0294.20035)], \textit{J. L. Brenner} and \textit{J. Wiegold} defined the spread of \(G,\) denoted \(s(G),\) to be the largest integer \(k\) such that for any nontrivial elements \(x_1,\dots,x_k\) in \(G,\) there exists \(y \in G\) such that \(G = \langle x_i, y\rangle\) for all \(i.\) This leads naturally to the more restrictive notion of uniform spread, denoted \(u(G)\): this is the largest integer \(k\) such that there is a conjugacy class \(C\) of \(G\) with the property that for any nontrivial elements \(x_1,\dots,x_k\) in \(G,\) there exists \(y\in C\) such that \(G = \langle x_i, y\rangle\) for all \(i.\) Clearly, \(s(G)\geq u(G)\) and \textit{T. Breuer} et al. [J. Algebra 320, No. 2, 443--494 (2008; Zbl 1181.20013)] proved that \(u(G) \geq 2\) for every non-abelian finite simple group \(G.\) For any group with \(u(G)\geq 1,\) the authors 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 \(G=\langle x, y\rangle.\) They establish several new interesting results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. In particular, they make substantial progress towards a classification of the simple groups \(G\) with \(\gamma_u(G) = 2,\) and they study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for \(G.\) They also establish new results concerning the 2-generation of soluble groups (in this case \(s(G), u(G)\) and \(\gamma_u(G)\) are completely determined) and symmetric groups (for example they proved that \(u(S_n)=2\) if \(n\neq 6\) and \(\gamma_u(S_n)\geq \log_2(n)\) for all \(n\)). Several interesting open problems are proposed.