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I have $$a= \left(\begin{matrix} 1 & 2 & 3 &4&5&6&7&8 \\ 1&3&8&7&6&5&2&4 \\ \end{matrix}\right) $$

My book gives no explanation as to how to write this as a product of 2 cycles. I can write it as a product of disjoint cycles. Help would be appreciated. Also, what is the significance of being either an odd or an even permutation?

Thanks in advance!

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  • $\begingroup$ $$a = (1)(2\,3\,8\,4\,7)(5\,6). $$ $\endgroup$ – Jack D'Aurizio Nov 3 '15 at 21:59
  • $\begingroup$ Can you write a cycle as a product of 2 cycles? If you can you can write the disjoint cycles, then expand each one into 2 cycles. Odd or even just tells you that you will have odd or even number of 2 cycles. $\endgroup$ – Paul Plummer Nov 3 '15 at 22:00
  • $\begingroup$ A completely different method: Your $a$ has only one fixpoint, $1$. Can you multiply it by a 2-cycle so that the result has two fix points, $1$ and $2$? Can you continue this scheme? What is the relation between $a$ and the sequence of 2-cycles thus picked? $\endgroup$ – Hagen von Eitzen Nov 3 '15 at 22:02
  • $\begingroup$ @PaulPlummer how do I do it tho? The book gives 2 cycles like 1 something, I do not know how to obtain them $\endgroup$ – Jennie Durham Nov 3 '15 at 22:04
  • $\begingroup$ Product of two cycles or product of 2-cycles (i.e., transpositions)? $\endgroup$ – lhf Nov 3 '15 at 22:32
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First, we have $a=(1)(23847)(56)$. The next thing to note is that if $\sigma=(i_1i_2\cdots i_n)$ is a cycle, we can express it as a product of transpositons as $$ \sigma=(i_1i_2)(i_2i_3)\cdots(i_{n-1}i_n). $$ In this example, you can verify that $$ (23847)=(23)(38)(84)(47). $$ Hence, $a=(23847)(56)=(23)(38)(84)(47)(56)$. It is a product of an odd number of transpositions, making it an odd permutation.

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    $\begingroup$ How do I write this cycle as a product of 1 something 2 cycles? Can you elaborate on that a little? $\endgroup$ – Jennie Durham Nov 3 '15 at 22:06
  • $\begingroup$ @JennieDurham I don't understand your question. You can write $(1)=(12)(12)$. Is that what you mean? $\endgroup$ – David Hill Nov 3 '15 at 22:55
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    $\begingroup$ Yes. I do not understand how something like $(23)$ can be written as $(12)(13)$ or $(1)=(12)(12)$? $\endgroup$ – Jennie Durham Nov 3 '15 at 22:59
  • $\begingroup$ First $(23)$ CANNOT be expressed as $(12)(13)=(132)$. You may want to get some help with cycle multiplication. It is a bit difficult to explain in a comment. $\endgroup$ – David Hill Nov 3 '15 at 23:06
  • $\begingroup$ Ok. Can you please check if $(678)(463)=(478)(63)$? I did some random calculation to see if I understand it $\endgroup$ – Jennie Durham Nov 3 '15 at 23:15

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