Sign of a permutation - sci.math

Message from discussion Sign of a permutation

William F. Hammond

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More options Apr 11 1990, 12:04 pm

Newsgroups: sci.math

From: wf...@leah.Albany.Edu (William F. Hammond)

Date: 11 Apr 90 16:04:27 GMT

Local: Wed, Apr 11 1990 12:04 pm

Subject: Re: Sign of a permutation

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In article <5...@ucrmath.UCR.EDU> b...@x.UUCP (john baez) writes:

>Help!!! I'd like a short proof if possible that the
>sign of a permutation is really well-defined, i.e.,
>that any element which is a product of an even number of
>interchanges is not the product of an odd number.
> . . .
>And can you do it by tomorrow morning????

>No, this is not a homework problem!!!!!!!!
>I'M TEACHING THE COURSE.

For a permutation in S_n define its sign to be 1 or -1 according to
what effect that permutation has on the polynomial

product for 1 <= i < j <=n of (x_i - x_j)

It is then obvious that the sign defined this way is a character, i.e.,
a homomorphism from S_n to the multiplicative group {1, -1}, that
assigns the value -1 to any transposition and, therefore, "counts mod 2"
the number of transpositions in any factorization of a permutation.
-- Bill

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