arXiv subjects

Piotr Słanina

Publications and source records attributed to Piotr Słanina.

2 recordsLinked to original sources

arXiv2020

On groups with weak Sierpiński subsets

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, namely of the form $G_k=\langle g,h\mid (h^{-1}g)^k\rangle$ unless it is free over $(g,h)$. In addition, in such groups $G_k$, we characterize all weak Sierpiński subsets.

math.GR↗

arXiv2019

On weak Sierpiński sets in groups and free subgroups

In this paper we discuss the problem of existence of so called weak Sierpiński sets in groups. It is known that group $G$ has a Sierpiński subset if and only if it contains a free subgroup. In their paper, Tomkowicz and Wagon conjectured that an analogous result holds also for the weaker condition. We derive a number of properties of groups with weak Sierpiński subsets and use them to prove the above mentioned conjecture. This is an improved version of arXiv:1805.11486. Some of the included proofs have been simplified and shortened.

math.GR↗