# 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.

Parents:

• Symmetric group

The symmetric groups form the fundamental link between group theory and the notion of symmetry.