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A subset SS of a group GG is called a total dominating set of GG if for any nontrivial element x∈Gx∈G there is an element y∈Sy∈S such that G=⟨x,y⟩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.