roxy potter
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Report Thread starter 13 years ago
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I've noticed on past exam papers at college we've been asked to state(not prove) a theorom justifying the uniqueness of the definitions of even and odd. I can't seem to find one. Does anybody know one?
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mikesgt2
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The theorem states that no permutation can be expressed as the product of an odd and even number of transpostions.

That is, if a permutation can be expressed as the product of an even number of transpositions, then it cannot be expressed as the product of an odd number of transpositions.

This is not trivial to prove but i can give you a proof if you'd like.
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mikesgt2
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Actually, if you need a proof follow question 6 in

http://www.maths.ox.ac.uk/current-st...ts-groups4.pdf

It leads you through a method of proving that the identity permutation cannot be expressed as the product of an odd number of permutations, which you should be able to see is equivalent to the required result.
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roxy potter
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Report Thread starter 13 years ago
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You're grand. The statement is all I need. When I get a chance I'll look it up, but my summer exams start next week and I'm trying to condense everything down. Thanks very much for you're help
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