Sign homomorphism on the finitary symmetric group
Definition for a finite set
Suppose is a finite set of size at least two. The sign homomorphism, also called the sign map or signum, on the symmetric group is a homomorphism to the group under multiplication, defined as follows. Without loss of generality, assume where . For a permutation , the sign of is given by:
For and , the sign homomorphism is defined by fiat to be the trivial map.
The definition is valid because:
- The map does go to : Both the numerator an denominator are products of the same linear factors, up to a difference of sign.
- The map is a homomorphism: This is easy to check, and follows essentially by cancellation of the denominator in one expression with the numerator in the next.
Further, any transposition (i.e., a permutation that interchanges two elements of and leaves everything else fixed) has sign . Thus, if a permutation is expressible as aproduct of transpositions, its sign is .
A permutation whose sign is is termed an even permutation and a permutation whose sign is is termed an odd permutation. The even permutations form the kernel of the sign homomorphism, and this subgroup of the symmetric group is termed the alternating group.