The Sign
Another important fact exhibits a normal subgroup An of
Sn . In order to show this we introduce the sign
e( p)
as follows:
e( p):= Õ1 <= i<j <= n( pj- pi)/(j-i) Î Z,
if
n >= 2, while e(1S 0):= e(1S
1):=1 Z.
As i not =j implies pi not = pj, we have e( p) not =0. Moreover,
the following sets of pairs are equal:
{ {i,j } | 1 £i<j £n }= { { pi, pj } | 1 £i<j £n },
and so
we have e( p)
= ±1 Z. Furthermore e is a homomorphism of Sn into
{1,-1 }:
e( pr)= Õi<j( prj- pri)/(j-i)= Õi<j
( prj- pri)/( rj- ri) Õi<j( rj- ri)/(j-i)= e( p) e( r).
This proves
Corollary:
The sign map
e:Sn -> {1,-1 } :p -> e( p)
is a homomorphism which is surjective for each n ³2.
Hence its kernel
An := kere= { pÎSn | e( p)=1 }
is a normal subgroup of Sn :
An lefttriangleeq Sn , | An | = | Sn | /2= n!/2, if n ³2.
The elements of An are called even
permutations, while the elements of Sn \An are called
odd
permutations.
Correspondingly, an r-cycle is even
if and only if r is odd.
There is a program to compute the
sign of various
permutations.
harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001
