Let \(n\geq 1\). The sign homomorphism is the map \[ \sgn\colon S_n \to (\ZZ/2, +), \qquad \sigma \mapsto k \bmod 2, \] where \(\sigma = \tau_1 \cdots \tau_k\) is any expression of \(\sigma\) as a product of \(k\geq 0\) transpositions.
Let \(n \geq 1\). For \(\sigma\in S_n\), all expressions of \(\sigma\) as a product of transpositions have the same number of factors modulo \(2\), so \(\sgn\) is well defined, and \(\sgn\) is a group homomorphism.