On certain permutation representations of the braid group. Part II

TL;DR

This paper characterizes when certain finite permutation representations of the braid group, which extend symmetric groups by abelian groups, are split extensions, building on previous work identifying these images.

Contribution

It provides a complete characterization of split extensions among specific finite permutation representations of the braid group.

Findings

Identified conditions under which these extensions are split.

Extended previous results on permutation representations of the braid group.

Clarified the structure of certain finite homomorphic images.

Abstract

In arXiv:0910.1727 we find certain finite homomorphic images of Artin braid group into appropriate symmetric groups, which a posteriori are extensions of the symmetric group on n letters by an abelian group. The main theorem of this paper characterizes completely the extensions of this type that are split.