Let \(n\geq 1\). A permutation \(\sigma\in S_n\) is even if \(\sgn(\sigma) = 0\), where \(\sgn\colon S_n\to\ZZ/2\) is the sign homomorphism. The alternating group is the subgroup of even permutations of the symmetric group: \[ A_n \coloneqq \theset{\sigma \in S_n \suchthat \sgn(\sigma) = 0} = \ker(\sgn). \]
A cycle of length \(\ell\) is a product of \(\ell-1\) transpositions, so a permutation is even if and only if its disjoint cycle decomposition has an even number of cycles of even length.