Parity of a Permutation

The parity of a Permutation σ : X ≃ X : 𝜎 similar-to-or-equals 𝑋 𝑋 \sigma:X\simeq X on a Finite set X 𝑋 X is the number of transpositions modulo 2. This induces a Group Homomorphism S n → ℤ / 2 ⁢ ℤ → subscript 𝑆 𝑛 ℤ 2 ℤ S_{n}\to\mathbb{Z}/{2}\mathbb{Z} .