Alternating group

Let \(n\geq 1\) and let \(\sgn\colon S_n\to\theset{\pm1}\) be the sign homomorphism. The alternating group is \[ A_n\coloneqq\ker(\sgn)=\theset{\sigma\in S_n \st \sgn(\sigma)=1}, \] the subgroup of even permutations of \(S_n\).