# The Chain Group of a Forest

## TL;DR

This paper introduces the concept of the chain group of a forest, exploring its properties and characterizing which groups can or cannot be realized as chain groups of forests.

## Contribution

It defines the chain group of a forest and characterizes its structure, including identifying which groups can be realized as chain groups and which cannot.

## Key findings

- Determined the chain groups for several families of forests.
- Proved that the dihedral group cannot be realized as a chain group of any forest.

## Abstract

For every labeled forest $\mathsf{F}$ with set of vertices $[n]$ we can consider the subgroup $G$ of the symmetric group $S_n$ that is generated by all the cycles determined by all maximal paths of $\mathsf{F}$. We say that $G$ is the chain group of the forest $\mathsf{F}$. In this paper we study the relation between a forest and its chain group. In particular, we find the chain groups of the members of several families of forests. Finally, we prove that no copy of the dihedral group of cardinality $2n$ inside $S_n$ can be achieved as the chain group of any forest.