# On groups in which Engel sinks are cyclic

## TL;DR

This paper investigates the structure of groups where each element's Engel sink is procyclic, proving the conjecture that such groups are procyclic-by-locally nilpotent in specific cases like finite and soluble pro-p groups.

## Contribution

It proves the conjecture for finite groups and soluble pro-p groups, advancing understanding of group structures with cyclic Engel sinks.

## Key findings

- Finite groups with cyclic Engel sinks are procyclic-by-nilpotent.
- Soluble pro-p groups with this property are also procyclic-by-nilpotent.

## Abstract

For an element $g$ of a group $G$, an Engel sink is a subset $\mathcal{E}(g)$ such that for every $ x\in G $ all sufficiently long commutators $ [x,g,g,\ldots,g] $ belong to $\mathcal{E}(g)$. We conjecture that if $G$ is a profinite group in which every element admits a sink that is a procyclic subgroup, then $G$ is procyclic-by-(locally nilpotent). We prove the conjecture in two cases -- when $G$ is a finite group, or a soluble pro-$p$ group.