How to write a cycle as a product of 2 cycles?

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?

• $$a = (1)(2\,3\,8\,4\,7)(5\,6).$$ – Jack D'Aurizio Nov 3 '15 at 21:59
• 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. – Paul Plummer Nov 3 '15 at 22:00
• 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? – Hagen von Eitzen Nov 3 '15 at 22:02
• @PaulPlummer how do I do it tho? The book gives 2 cycles like 1 something, I do not know how to obtain them – Jennie Durham Nov 3 '15 at 22:04
• Product of two cycles or product of 2-cycles (i.e., transpositions)? – lhf Nov 3 '15 at 22:32

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.
• @JennieDurham I don't understand your question. You can write $(1)=(12)(12)$. Is that what you mean? – David Hill Nov 3 '15 at 22:55
• Yes. I do not understand how something like $(23)$ can be written as $(12)(13)$ or $(1)=(12)(12)$? – Jennie Durham Nov 3 '15 at 22:59
• 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. – David Hill Nov 3 '15 at 23:06
• Ok. Can you please check if $(678)(463)=(478)(63)$? I did some random calculation to see if I understand it – Jennie Durham Nov 3 '15 at 23:15