On the Total Dominating Set of ${3}/{2}$-Generated Groups - Atabekyan, Varujan; Aslanyan, Haik (Armenian Journal of Mathematics, 2024)
A subset $S$ of a group $G$ 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 =\left< x, y\right> $. Tarski monsters, constructed by Olshanskii, are infinite simple groups, any pair of non-commuting elements of which is a total dominating set. In this paper, we construct an infinite non-cyclic and non-simple group having a total dominating set from two elements. This gives a positive answer to Donoven and Harper's question about the existence of infinite groups (other than Tarski monsters) having a finite total dominating set. In addition, our examples have an infinite uniform spread.
Keywords: Spred of Group; Total Dominating Set; 3/2-generated Group; Tarski Monster