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?
How to prove that any 4-cycle can be written as the product of simple 4-cycles
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$\begingroup$
abstract-algebra
group-theory
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0What is a "simple" $4$-cycle? What other kinds of $4$-cycles are there? – 2018-10-15
1 Answers
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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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0@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. – 2011-03-23