Let $\sigma \in S_n$ s.t $\sigma = (1 ... n)$. Prove $C_{S_n}(\sigma) = \langle\sigma\rangle$.
So we this is what I have and I'll also explain where I'm stuck:
We know $C_{S_n}(\sigma) = \{\tau \in S_n | \tau\sigma\tau^{-1} = \sigma\} = \{\tau \in S_n | (\tau(1) ... \tau(n)) = (1 ... n)\}$.
This means $\tau \in S_n$ satisfices $\tau \in C_{S_n}(\sigma) \iff \tau$ is a cyclic shift of $\sigma$.
The trouble I'm having is that $\langle \sigma \rangle$ doesn't satsify this assumption because for example (if n is even WLOG) :$(1 ... n)(1 ... n) = (1\;3 ... n-1)(2\;4 ... n)$ which isn't a cyclic shift.