# Proving the sign map is a homomorphism

Definition: A transposition is a 2-cycle permutation.

Definition: A permutation $$\sigma$$ is called even if its decomposition into transpositions has even number of transpositions; analogously for odd permutations.

Now, coming to the question, how do I prove using this definition that the sign map $$\operatorname{sgn}\colon S_n\to\{\pm 1\}$$ assigning a permutation in the symmetric group $$S_n$$ to its sign (+1 if even, -1 if odd) is a homomorphism?

I know how to prove it using the definition of the sign map being the determinant of the $$n\times n$$ permutation matrix associated to a permutation (in which case it is obvious since $$\operatorname{det}$$ is multiplicative).

Then if $$\sigma=t_1...t_n, sign(\sigma)=(-1)^n$$ this is well defined since the parity of $$n$$ does not depend of the composition. If $$\sigma'=t'_1....t'_{n'}, sign(\sigma')=(-1)^{n'}$$ and as $$\sigma\sigma'=t_1....t_nt'_1...t'_{n'}, sign(\sigma\sigma')=(-1)^{n+n'}$$.