On groups with weak Sierpiński subsets
Volume 252 / 2021
Fundamenta Mathematicae 252 (2021), 171-178
MSC: Primary 20F05; Secondary 20E05.
DOI: 10.4064/fm681-2-2020
Published online: 3 August 2020
Abstract
In a group $G$, a weak Sierpiński subset is a subset $E$ such that for some $g,h\in G$ and $a\neq b\in E$, we have $gE=E\smallsetminus \{a\}$ and $hE=E\smallsetminus \{b\}$. In this setting, we study the subgroup generated by $g$ and $h$, and show that it has a special presentation, $G_k=\langle g,h\mid (h^{-1}g)^k\rangle $, unless it is free over $g$ and $h$. In addition, in such groups $G_k$, we characterize all weak Sierpiński subsets.
