TY - JOUR
TI - On the Total Dominating Set of ${3}/{2}$-Generated Groups
PY - 2024/10/01
Y2 - 2024/11/12
JF - Armenian Journal of Mathematics
JA - Armen.J.Math.
VL - 16
IS - 10
SE - Articles
DO - 10.52737/18291163-2024.16.10-1-7
UR - https://doi.org/10.52737/18291163-2024.16.10-1-7
SP - 1-7
AB - A subset $S$ of a group $G$ is called a total dominating set of $G$ if for any nontrivial element $x\in G$ there is an element $y\in S$ such that $G =\left< x, y\right> $. 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.
ER -