On certain permutation representations of the braid group

TL;DR

This paper proves a structural theorem about specific permutation representations of the braid group, showing they are extensions of symmetric groups by abelian groups, with some cases splitting.

Contribution

It introduces a new structural understanding of certain homomorphic images of the braid group in symmetric groups, including conditions for splitting extensions.

Findings

Permutation groups are extensions of symmetric groups by abelian groups.

In half of the cases, these extensions split.

Provides a classification of these permutation representations.

Abstract

This paper is devoted to the proof of a structural theorem, concerning certain homomorphic images of Artin braid group on $n$ strands in finite symmetric groups. It is shown that any one of these permutation groups is an extension of the symmetric group on $n$ letters by an appropriate abelian group, and in "half" of the cases this extension splits.