On the Total Dominating Set of 3 / 2 -Generated Groups

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On the Total Dominating Set of 3 / 2 -Generated Groups Varujan Atabekyan

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.

DOI: 10.52737/18291163-2024.16.10-1-7 Armenian Journal of Mathematics, 16(10) 1-7

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Keywords

Spred of Group, Total Dominating Set, 3/2-generated Group, Tarski Monster

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