Armenian Journal of Mathematics, 16(10)
2024; 1–7
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On the Total Dominating Set of 3 / 2 -Generated Groups

Varujan Atabekyan*, Haik Aslanyan

Received: 2024-12-03 · Published: 2024-10-01

© 2024 The Author(s) and the original publisher.

CC BY 4.0 This work is licensed under a Creative Commons Attribution 4.0 International License.
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Original title
On the Total Dominating Set of 3 / 2 -Generated Groups
Authors
Varujan Atabekyan, Haik Aslanyan
Source journal
Armenian Journal of Mathematics, 16(10)
Published
2024-10-01
Licence
Creative Commons Attribution 4.0 International
Original
https://doi.org/10.52737/18291163-2024.16.10-1-7

Abstract

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.
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