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:

$⟨s_1,s_2,\dots ,s_{n-1}\mid s_i{s}_{i+1}{s}_i=s_{i+1}{s}_i{s}_{i+1}\mathrm{\forall }1\le i\le n-2,s_i{s}_j=s_j{s}_i\mathrm{\forall }|i-j|>1⟩$.

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