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:

$\langle s_{1},s_{2},\dots ,s_{n-1}\mid s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}\ \forall \ 1\leq i\leq n-2,s_{i}s_{j}=s_{j}s_{i}\ \forall \ |i-j|>1\rangle$.

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