Small Quotients of Braid Groups

TL;DR

This paper investigates the minimal non-cyclic quotients of braid groups, establishing specific smallest quotients for certain n and providing improved bounds on quotient orders.

Contribution

It identifies the smallest non-cyclic quotients of braid groups for n=5,6 and the smallest non-trivial quotients of their commutator subgroups for n=5 to 8, with improved bounds.

Findings

S_n is the smallest non-cyclic quotient of B_n for n=5,6

A_n is the smallest non-trivial quotient of B_n' for n=5,6,7,8

Provides an improved lower bound on the order of non-cyclic quotients of B_n

Abstract

We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is the smallest non-trivial quotient of the commutator subgroup $B_n'$ for $n = 5,6,7,8$. We also give an improved lower bound on the order of any non-cyclic quotient of $B_n$.