TITLE: Let $σ∈S_n$ be a $n$ cycle. Prove that an element $τ∈S_n$ only commutes with $σ$ if $τ$ is a power of $σ$.
QUESTION [12 upvotes]: Let $σ∈S_n$ be a $n$ cycle. Prove that an element $τ∈S_n$ only commutes with $σ$ if $τ$ is a power of $σ$.

REPLY [12 votes]: Suppose WLOG
$$
\sigma = (1 2 3 \dots n).
$$
Then
$$
(1 2 3 \dots n) = \tau (1 2 3 \dots n) \tau^{-1} = (\tau(1) \tau(2) \dots \tau(n)).
$$
So if $\tau(1) = i+1$ (for some $0 \le i < n$), then $\tau(2) = i+2$, etc  , $\tau(j) = j+i$ (Kasper correctly notes - thanks! - in a comment that once we get to $n$, we should fold over to $1$, essentially arguing modulo $n$), so $\tau = \sigma^{i}$.