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So I am still getting the hang of cyclic notation.

Express the following permutations as products of transpositions and identify them as even or odd. I think this is saying express the following in pairs? like (xx), so my attempt:

a. (14356)=(61)(56)(35)(43)(14)

b. (156)(234)=(61)(56)(15)(42)(34)(23)

c. (1426)(142)=(61)(26)(42)(14)

My guess is that they're all even because I wrote them all as an even number of pairs? Not sure if this is even the correct reasoning or if my answers are right. Any guidance would be much appreciated!

Math Major
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  • You are starting off well by considering the disjoint cycle factors. But then things get off track. E.g. multiply the product $(61)(56)(15)$ from b. back to a single permutation. What do you get? It isn't $(156)$ as you expected it to be. – hardmath Mar 06 '15 at 01:08
  • All factorisations are wrong. Make sure you read them right-to-left. – AlexR Mar 06 '15 at 01:08

1 Answers1

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Here are two simple things you might prove:

$$(a_1,a_2,...,a_n) = (a_1,a_2)(a_2,a_3)(a_3,a_4)...(a_{n-1},a_n)$$

$$(a_1,a_2,...,a_n) = (a_1,a_n)(a_1,a_{n-1})(a_1,a_{n-2})...(a_1,a_2)$$

  • This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. – kjetil b halvorsen Mar 06 '15 at 01:40
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    "Any guidance would be much appreciated!". This answers how to perform the decomposition, such that they can check their work. –  Mar 06 '15 at 01:42