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Here I asked how I can write a particular 4-cycle as a product of simple 4-cycles and I understand the solutions given. I now want to prove that every 4-cycle can be written as the product of simple 4-cycles. The only way that I know of to prove this is induction. I have no problem with induction but my problem is that I can not come up with a good statement for induction that I can prove. So can you please give me some hints on this?

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It isn't true in $S_4$. – Jonas Meyer Mar 23 '11 at 7:03
What about $S_n$ when $n\geq5$? – Vafa Khalighi Mar 23 '11 at 7:12

For $n \ge 5$, it is not long to see that $4$-cycles generate $S_n$, so what you want to show is equivalent to showing that simple $4$-cycles generate $S_n$.

If you prove that simple $4$-cycles generate $S_n$, then since $S_n$ and the simple $4$-cycle $(n-2, n-1, n, n+1)$ generate $S_{n+1}$, they generate $S_{n+1}$.

So you only need to prove that simple $4$-cycles generate $S_5$, and a simple finite computation can show that they do.

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That is my main weakness with permutations. I have no idea how I could write $(1,2,3,4,5)$ as a product of simple 4-cycles or even just normal 4-cycles! – Vafa Khalighi Mar 23 '11 at 10:18
@Vafa Khalighi : Well it's just a matter of writing every possible product combination of $a = (1,2,3,4)$ and $b = (2,3,4,5)$ until you find everything you wanted to find. I got that $(1,2,3,4,5) = aababa$ and $(1,2) = ababb$ for example. – mercio Mar 23 '11 at 12:08

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