Sign homomorphism (from the symmetric group)

The sign homomorphism is given by sending a permutation $\sigma$ in the Symmetric group $S_n$ to $0$ if we can make $\sigma$ by multiplying together an even number of transpositions, and to $1$ otherwise.

%%%knows-requisite(Modular arithmetic): Equivalently, it is given by sending $\sigma$ to the number of transpositions making it up, modulo $2$. %%%

The sign homomorphism is well-defined.

%%%knows-requisite(Quotient group): The Alternating group is obtained by taking the quotient of the symmetric group by the sign homomorphism. %%%