A subset S of a group G is called a total dominating set of G if for any nontrivial element x∈ G there is an element y∈ S 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.
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Atabekyan et al. (2024) studied this question.
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