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I'm trying to go through the proof that a subgroup of $\mathbb{S}_5$ with a 5-cycle and a transposition is the whole group found on this link. However, I'm not able to understand why we can assume that the 5-cycle is $(1,2,3,4,5)$. This is not the only reference where I saw it assumed, but I still don't see clearly why is this the case. How can I get that 5-cycle from any 5-cycle?

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  • $\begingroup$ I mean, if all you really want is the assumption that the $5$-cycle is $(1,2,3,4,5)$, you can use the handy fact that mathematics does not depend on symbols used. :P The link goes a bit further in that it assumes simultaneously that the $5$-cycle is $(1,2,3,4,5)$ and the transposition is $(1,i)$. Matt Samuel's answer shows a kind of autmorphism that should take care of that. $\endgroup$ – Fimpellizieri Dec 7 '16 at 4:32
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$$\sigma(1,2,3,4,5)\sigma^{-1}=(\sigma(1),\sigma(2),\sigma(3),\sigma(4),\sigma(5))$$

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