Sign Homomorphism
The map sends a product of transpositions to and is a well-defined homomorphism.
Theorem 8.15 (Sign Homomorphism)
Proof
If
then . The parity lemma makes even, so and the sign is well-defined. Concatenating transposition factorisations gives
so is a homomorphism.
Related
Stated in
- Theorem 8.15 (Sign Homomorphism)§8.2 Transpositions and the Sign Homomorphism