# On groups with weak Sierpi\'nski subsets

## TL;DR

This paper investigates groups with weak Sierpiński subsets, characterizing the structure of subgroups generated by specific elements and describing all such subsets within certain groups.

## Contribution

It introduces a new class of subsets in groups, analyzes the subgroup structures they generate, and characterizes all weak Sierpiński subsets in particular groups.

## Key findings

- Subgroups generated by specific elements have a special presentation.
- Weak Sierpiński subsets are characterized in groups with certain subgroup structures.
- Most such groups are either of a specific presentation or free over generators.

## Abstract

In a group $G$, a weak Sierpi\'nski 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 Sierpi\'nski subsets.