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I had to find two 5-cycles in $A_5$ that are not conjugate. I believe I found two, namely $a=(12345)$ and $b=(21345)$.

It wasn't part of the exercise, but I'd like to prove that they are not conjugate. But I'm not sure where to start. I think we have to start with that they are conjugate in $S_5$, so there exists a $g \in S_5$ such that $gag^{-1}=b$. And then prove all the $g's$ for which that is true are $\notin A_5$?

Someone stated this as a duplicate, but I don't think it is, because they don't really ask for/ give the algebraic proof that I'm looking for.

  • Two cycles with the same length are always conjugate. – Bernard Oct 10 '17 at 11:34
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    @Bernard They are not conjugate, because there are 4!=24 5-cycles in $A_5$, the order of $A_5$ is 60, and 24 does not divide 60. –  Oct 10 '17 at 11:39

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