On the total dominating set of 3/2-generated groups (Q6643145)

Let \(G\) be a \(2\)-generated group. The spread of \(G\), denoted by \(s(G)\) is the greatest \(k\) such that for any non-trivial elements \(x_{1}, \ldots x_{k} \in G\) there exists \(y \in G\) such that \(G = \langle x_{i}, y \rangle\) for all \(i = 1, \ldots, k\). The uniform spread \(u(G)\) of \(G\) is the greatest \(k\) such that there is a conjugacy class \(C \subseteq G\) with the property that for any nontrivial elements \(x_{1}, \ldots, x_{k}\) there exists \(y \in C\) such that \(G= \langle x_{i}, y\rangle\) for all \(i = 1, \ldots, k\).\N\NA subset \(S\) of \(G\) is called a total dominating set of \(G\) if for any nontrivial element \(x\in G\) there is an element \(y \in S\) such that \(G =\langle x, y \rangle\).\N\NIn the paper under review, the authors construct an infinite non-cyclic and non-simple group having a total dominating set \(S\) with \(|S|=2\). This gives a positive answer to a question by \textit{C. Donoven} and \textit{S. Harper} [Bull. Lond. Math. Soc. 52, No. 4, 657--673 (2020; Zbl 1472.20069)] about the existence of infinite groups (other than the so-called Tarski monsters) having a finite total dominating set. Furthermore, such examples have infinite uniform spread.