Any 5-cycle in $\mathbb{S}_5$ can be obtained from another 5-cycle

Question

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?

Answer 1

$$\sigma(1,2,3,4,5)\sigma^{-1}=(\sigma(1),\sigma(2),\sigma(3),\sigma(4),\sigma(5))$$