 Pure braid group

The pure braid group $P_n$ is the kernel of the endpoint-permutation homomorphism from the braid group $B_n$ to the symmetric group $S_n$:Each strand of a pure braid returns to its own original endpoint. It may still wind around other strands, so trivial endpoint permutation does not make the braid trivial.

For two strands, $B_2\cong \mathbb{Z}$ is generated by one crossing ${\sigma }_1$, and $P_2$ consists of its even powers. Thus $P_2\cong \mathbb{Z}$ but has index two in $B_2$. In general, the endpoint map is onto, so $[B_n:P_n]=n!$.

Geometrically, $P_n$ is the fundamental group of the configuration space of $n$ ordered distinct points in the plane. The ordering records which moving point must return to which starting point, matching the pure-strand condition.