The Sign |

Another important fact exhibits a normal subgroup * A _{n} * of

e( p):= Õ_{1 <= i<j <= n}(pj- pi)/(j-i)ÎZ, 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( 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 mapis a homomorphism which is surjective for eache:S_{n}-> {1,-1 } :p -> e( p)n ³2. Hence its kernelis a normal subgroup ofA_{n}:= kere= { pÎS_{n}| e( p)=1 }S:_{n}A_{n}lefttriangleeq S_{n}, | A_{n}| = | S_{n}| /2= n!/2, if n ³2.

The elements of * A _{n} * are called

permutations, while the elements of * S _{n} \A_{n} * are called

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 "at" uni-graz.at | http://www-ang.kfunigraz.ac.at/~fripert/ |

UNI-Graz | Institut für Mathematik |

UNI-Bayreuth | Lehrstuhl II für Mathematik |

The Sign |