# 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.