Braid group

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 $$n!$$.