Message from discussion Sign of a permutation

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From: v088k...@ubvmsd.cc.buffalo.edu (James D Dolan)
Newsgroups: sci.math
Subject: Re: Sign of a permutation
Message-ID: <21905@eerie.acsu.Buffalo.EDU>
Date: 16 Apr 90 04:56:56 GMT
References: <9580@sdcc6.ucsd.edu> <Aa88F1m00Ws9Q3tApd@andrew.cmu.edu> <5434@ucrmath.UCR.EDU> <826@s6.Morgan.COM>
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Posted: Mon Apr 16 05:56:56 1990

In article <5...@ucrmath.UCR.EDU>, b...@x.ucr.edu (john baez) writes...
>the "intrinsic" proofs that the identity cannot be
>written as an odd number of transpositions seem rather  
>clunky, though they have a kind of "homebrew" charm of

I DON'T SEE WHY THE "INTRINSIC" APPROACH HAS TO BE PRESENTED IN A CLUNKY
WAY. HERE IS MY ATTEMPT AT A NON-CLUNKY PRESENTATION:

DEFINE THE SIGN OF A PERMUTATION AS THE "NORMALIZED" PARITY OF THE NUMBER OF
CYCLES ("NORMALIZED" IN THE SENSE THAT THE IDENTITY PERMUTATION IS ALWAYS
TAKEN TO HAVE EVEN PARITY). WELL-DEFINEDNESS IS THEN TRIVIAL, AND THE
HOMOMORPHISM PROPERTY FOLLOWS FROM THE FACT THAT EVERY PERMUTATION IS A
PRODUCT OF TRANSPOSITIONS, PLUS THE FOLLOWING:

LEMMA: MULTIPLICATION BY A TRANSPOSITION CHANGES THE PARITY OF THE NUMBER OF
CYCLES.
PROOF: MULTIPLICATION BY A TRANSPOSITION SPLICES TWO CYCLES INTO ONE IF THE
TRANSPOSED ELEMENTS ARE IN DIFFERENT CYCLES, AND SLICES ONE CYCLE INTO TWO IF
THEY ARE IN THE SAME CYCLE.


JAMES DOLAN, MATH DEPT., SUNY AT BUFFALO