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For $n\ge 1$, the braid group $B_n$ consists geometrically of braids on $n$ strands, composed by stacking. Algebraically it is generated by ${\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to the braid group relations: adjacent generators obey ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$, and nonadjacent generators commute. There is no relation ${\sigma }_i^2=1$.

Recording only the strand endpoints gives a surjective group homomorphism $B_n\to S_n$, sending ${\sigma }_i$ to the transposition $(ii+1)$. Its group kernel is the pure braid group, whose strands return to their original endpoints. Thus endpoint permutations discard crossing information. For example, $B_2$ has one generator and no relations, so it is an infinite cyclic group, while its endpoint permutation group is $S_2$; the square of its generator has trivial endpoint permutation but is a nontrivial braid. This distinction does not conflict with Cayley's theorem, which realizes every group as some permutation group.