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I have a cycle here that I have broken down into 2 disjoint cycles, them being (1,5,6) and (2,8). I'm wondering what is the process of telling whether the set is even or odd. Is it the number of disjoint cycles or the size?

Cycle image

Arthur
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Ceolan Mc Mullin
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1 Answers1

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A cycle of even length / order is an odd permutation, and vice versa (that's just a conventional dissonance we have to live with; the alternative is worse). The product of two odd or of two even permutations (not just cycles) is even, while the product of one even and one odd permutation (in whatever order) is odd. The identity permutation is considered even.

Armed with this knowledge, we get that the cycle $(1,5,6)$ has length $3$ and is therefore even, and $(2,8)$ has length $2$ and is therefore odd, so their product is odd.

Arthur
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  • Right, I think I understand, and if there is more than one cycle. Say if there were 2 even and 1 odd (i.e 2 of length 3 and 1 of length 2) that would be an odd permutation also? – Ceolan Mc Mullin Dec 18 '19 at 16:09
  • @CeolanMcMullin Yup. An odd number of odd permutations (and any number of even permutations) makes an odd permutation, otherwise it's even. Also if they aren't disjoint. – Arthur Dec 18 '19 at 16:10
  • So would I be correct in saying the 3 rules are:
    1. if the 2 permutations/not all permutations are odd or even, it will be odd
    2. any number of even permutations will lead to the cycle being odd
    3. an odd number of odd permutations will be odd
     – Ceolan Mc Mullin Dec 18 '19 at 16:21
  • @CeolanMcMullin No, it's simpler. It's just rule 3). And any permutation that isn't odd according to rule 3) is even. – Arthur Dec 18 '19 at 17:05