Another important fact exhibits a normal subgroup 
of
. In order to show this we introduce the sign
as follows:
As 
implies 
we have 
Moreover,
the following sets of pairs are equal:
and so
we have 
. Furthermore 
is a homomorphism of into
:
This proves
is a homomorphism which is surjective for each
is a normal subgroup of
.
Corollary
The sign map
.
Hence its kernel
:
The elements of are called even
permutations, while the elements of 
are called
odd
permutations.
Correspondingly, an 
-cycle is even
if and only if is odd.
There is a program to compute the
sign of various
permutations.