Braid groups naturally give rise to a surjective group homomorphism \(\gamma : \B_n \to S_n\).
Let \(\beta\) be a \(n\)-braid, given that strands connect points \(P_i\) to \(Q_{\pi(i)}\), we define a homomorphism \(\gamma : \B_n \to S_n\) such that \[ \gamma(\beta) = \begin{pmatrix} 1 & \cdots & i & \cdots & n \\ \pi(1) & \cdots & \pi(i) & \cdots & \pi(n) \end{pmatrix} \] This homomorphism is in essence the result of disregarding how the strands cross.
By “disjoint” elementary braids and disjoint cycles commuting, the image of composition of braids is the same as the composition of images of braids.
Also similar to the symmetric groups, the braid groups can be easily coerced into larger groups, i.e. there is a natural way to fit \(\B_n\) into \(\B_{n+1}\). In both cases additional “elements” may be included.
For instance, a cycle representation of a permutation on \(n\) letters in the symmetric group can be applied to a set of \(m\) letters, \(m \gt n\), simply by considering the unlisted numbers as being within their own cycle. For example, the cycle \((1\,3\,2)\) representing a permutation of three letters can also represent a permutation on four letters as in \(S_4\), \((1\,2\,3) = (1\,2\,3)(4)\). Similarly, consider an arbitrary \(n\)-braid, and then add a single trivial strand connecting points \(P_{n+1}\) and \(Q_{n+1}\). In both cases there is a natural way to expand elements to elements of the larger group.