On the Total Dominating Set of ${3}/{2}$-Generated Groups
DOI:
https://doi.org/10.52737/18291163-2024.16.10-1-7Keywords:
Spred of Group, Total Dominating Set, 3/2-generated Group, Tarski MonsterAbstract
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.
References
J.L. Brenner and J. Wiegold, Two-generator groups. I. Michigan Math. J., 22 (1975), no. 1, pp. 53-64.
T. Breuer, R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups, II. J. Algebra, 320 (2008), no. 2, pp. 443-494.
T.C. Burness and S. Guest, On the uniform spread of almost simple linear groups. Nagoya Math. J., 209 (2013), pp. 35-109.
T. Burness, R. Guralnick and S. Harper, The spread of a finite group. Ann. of Math., 193 (2021), no. 2, pp. 619-687.
Ch.G. Cox, On the spread of infinite groups. Proceedings of the Edinburgh Mathematical Society, 65 (2022), pp. 214-228.
C. Donoven and S. Harper, Infinite 3/2-generated groups. Bull. London Math. Soc., 52 (2020), no. 4, pp. 657-673.
R.M. Guralnick and W.M. Kantor, Probabilistic generation of finite simple groups. J. Algebra, 234 (2000), no. 2, pp. 743-792.
R.M. Guralnick, E. Plotkin and A. Shalev, Burnside-type problems related to solvability. Internat. J. Algebra Comput., 17 (2007), no. 5-6, pp. 1033-1048.
A.Yu. Olshanskii, Groups of bounded period with subgroups of prime order. Algebra Logika, 21 (1982), no. 5, pp. 553-618.
A.Yu. Olshanskii, The Geometry of Defining Relations in Groups, Kluwer Academic Publishers, 1991.
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