Parity_Function_is_Homomorphism_v2

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\map \sgn \pi$ be the sign of $\pi$. Let the parity function of $\pi$ be defined as Parity of $\pi = \begin{cases} \mathrm {Even} & : \map \sgn \pi = 1 \\ \mathrm {Odd} & : \map \sgn \pi = -1 \end{cases}$ The mapping $\sgn: S_n \to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism.