The natural homomorphism $B_n \to \Sigma_n$ sending $\sigma_i$ to a transposition

Every braid has an underlying permutation of its $n$ strands, and this assignment is a group homomorphism.

Let $B_n$ be the abstract braid group with Artin generators $\sigma_1,\dots,\sigma_{n-1}$ and defining relations

and let $\Sigma_n$ be the symmetric group on the strand labels $\{1,\dots,n\}$. The assertion is that there exists a group homomorphism

sending each elementary braid to the transposition of the two strands it interchanges. Equivalently, the transpositions $(i,i+1)$ satisfy Artin's two families of relations in $\Sigma_n$, so the assignment on generators extends to the whole group.

This homomorphism is the algebraic counterpart of the map that records, for a motion of $n$ points of the plane, the permutation by which the points are rearranged; its kernel is the pure braid group. It is the basic tool for separating a braid's combinatorial shadow from its genuinely braided content, and it is used whenever a statement about $B_n$ is reduced to the corresponding statement about pure braids.

Formalization Note Braid generators are indexed by $\mathrm{Fin}(n-1)$ with truncated subtraction, the index $i$ standing for the book's $\sigma_{i+1}$; strandIdx i and strandIdxSucc i are the two strand labels $i$ and $i+1$ in $\mathrm{Fin}\,n$ that the generator interchanges. The statement is existential because the homomorphism is being constructed, and it is pinned down uniquely by its values on the generators, which generate the group.