= Pure braid group
{title2=$P_n$}
The pure braid group $P_n$ is the kernel of the endpoint-permutation homomorphism from the <braid group> $B_n$ to the <symmetric group> $S_n$:
$$
1\longrightarrow P_n\longrightarrow B_n\longrightarrow S_n\longrightarrow1.
$$
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.
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