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 xG there is an element yS such that G =< x, y> . 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.