# Braid group

## Definition

### In terms of the presentation using Artin braid relations

The braid group on $n$ letters, denoted $B_n$, is defined as follows:

$\langle s_1, s_2, \dots, s_{n-1} \mid s_is_{i+1}s_i = s_{i+1}s_is_{i+1} \ \forall \ 1 \le i \le n - 2, s_is_j = s_js_i \ \forall \ |i - j| > 1 \rangle$.

## Facts

There is a natural surjective homomorphism from the braid group $B_n$ to the symmetric group $S_n$, that sends each $s_i$ to the transposition $(i,i+1)$ in $S_n$. One way of seeing this is noting that the presentation of $S_n$ is obtained by tacking on more relations (namely, the relations that each $s_i$ square to the identity) to the relations for $B_n$.

The kernel of this homomorphism is the pure braid group and is denoted $P_n$. $P_n$ is thus a normal subgroup of finite index in $B_n$. The index is $n!$.

## Particular cases

Value of $n$ Value of $n - 1$ (number of generators for the Artin presentation) Braid group $B_n$ Symmetric group $S_n$ Pure braid group $P_n$(kernel of natural homomorphism to symmetric group)
1 0 trivial group trivial group trivial group
2 1 group of integers cyclic group:Z2 group of integers
3 2 braid group:B3 symmetric group:S3 pure braid group:P3
4 3 braid group:B4 symmetric group:S4 pure braid group:P4
5 4 braid group:B5 symmetric group:S5 pure braid group:P5