- #1
MostlyHarmless
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I'm asked to show that a permutation is even if and only if the number of cycles of even length is even. (And also the odd case)
I'm having trouble getting started on this proof because the only definitions of parity of a permutation I can find are essentially this theorem. And obviously I can't use this theorem to prove this theorem.. (If only). So what is the most basic, abstract definition of parity of a permutation that I might use, for a permutation of a set of size, n, that is even.
And as a side note, I've not gotten my textbook yet, I've been mooching off my class mates because, well, books are expensive and I'm broke atm. So I'm sure a relevant definition for parity of a permutation is in there but I don't have access to it right now.
Edit: We've been using what we called double row notation for permutations I.e.
(1 2 3 4 5)
(2 4 3 5 1)
And this is the question. Statement up until "Show that.."
Let ##{\sigma}{\epsilon}S_n## and suppose that ##{\sigma}## can be written as a product of disjoint cycles. Show that..
I'm having trouble getting started on this proof because the only definitions of parity of a permutation I can find are essentially this theorem. And obviously I can't use this theorem to prove this theorem.. (If only). So what is the most basic, abstract definition of parity of a permutation that I might use, for a permutation of a set of size, n, that is even.
And as a side note, I've not gotten my textbook yet, I've been mooching off my class mates because, well, books are expensive and I'm broke atm. So I'm sure a relevant definition for parity of a permutation is in there but I don't have access to it right now.
Edit: We've been using what we called double row notation for permutations I.e.
(1 2 3 4 5)
(2 4 3 5 1)
And this is the question. Statement up until "Show that.."
Let ##{\sigma}{\epsilon}S_n## and suppose that ##{\sigma}## can be written as a product of disjoint cycles. Show that..
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