The Sign

The Sign

Another important fact exhibits a normal subgroup A_n of S_n . In order to show this we introduce the sign   e( p) as follows:

e( p):= Õ_{1 <= i<j <= n}( pj- pi)/(j-i) Î Z, if n >= 2, while e(1_{S 0}):= e(1_{S 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 S_n into {1,-1 }:

e( pr)= Õ_i<j( prj- pri)/(j-i)= Õ_i<j ( prj- pri)/( rj- ri) Õ_i<j( rj- ri)/(j-i)= e( p) e( r).

This proves

Corollary: The sign map

e:S_n -> {1,-1 } :p -> e( p)

is a homomorphism which is surjective for each n ³2. Hence its kernel

A_n := kere= { pÎS_n | e( p)=1 }

is a normal subgroup of S_n :

A_n lefttriangleeq S_n , | A_n | = | S_n | /2= n!/2, if n ³2.

The elements of A_n are called even  

permutations, while the elements of S_n \A_n 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.

The Sign