Sign of a permutation - sci.math

Message from discussion Sign of a permutation

Mark Steinberger

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More options Apr 11 1990, 2:21 pm

Newsgroups: sci.math

From: ms...@leah.Albany.Edu (Mark Steinberger)

Date: 11 Apr 90 18:21:23 GMT

Local: Wed, Apr 11 1990 2:21 pm

Subject: Re: Sign of a permutation

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In article <2...@leah.Albany.Edu> wf...@leah.albany.edu.UUCP (William F. Hammond) writes:

>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.
>> . . .

>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

The above proof is of course elegant. It also requires some
sophistication of the student, so it may not be the best one if the
students haven't been exposed to polynomial rings and groups of units.

If the course is concentrating on group theory at the moment, it may
be instructive to grind through a proof along the following lines:

1. Show that every permutation has a unique decomposition as a product
of disjoint cycles.

2. Define the sign of a k-cycle to be (-1)^{k-1} (i.e. (-1) to the
smallest number of transpositions one can multiply to obtain a
k-cycle).

3. Define the sign of a product of disjoint cycles to be the product
of the signs of the cycles in it.

4. Let sigma be any product of disjoint cycles and let tau be a
transposition. Explicitly write down the product tau sigma as a
product of disjoint cycles. (If tau permutes i and j, then the result
depends on how i and j sit in the cycle structure of sigma.)
By explicit calculation, show that the
sign of tau sigma is the negative of the sign of sigma.

A proof along these lines is a bit tedious, but totally elementary. It
also shows the students that if they take an idea, like the
decomposition of a permutation as a product of disjoint cycles, then
they can use it to straightforwardly prove a useful result, given
sufficient confidence and persistence.

--Mark

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