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. %%%