Sign homomorphism (from the symmetric group)

by Patrick Stevens Jun 17 2016

The sign homomorphism is how we extract the alternating group 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. %%%