Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$.
My answer Let $\sigma \in S_n$. If $$\sigma (1 \cdots n) = (1 \cdots n) \sigma$$ then $$\sigma (1 \cdots n) \sigma^{-1} = (1 \cdots n).$$ This implies $$( \sigma(1) \cdots \sigma(n)) = (1 \cdots n).$$ We conclude $\sigma = ()$, which is equal to $(1\cdots n)^i$ for $i$ that are multiples of $n$.
Problem This just seems really silly. Why would the question mention "certain $i$"? There are many ways to write $(1)$...
Say $n=3$ for example. Then $(\sigma(1)\sigma(2)\sigma(3))=(123)$. But that's not enough to conclude the equality $\sigma(k)=k$ for $k=1,2,3$. Why? Because $(123)$ can also be written $(231)$ and $(312)$, so we could also have other posisble $\sigma$s as well.
Hint: suppose $\sigma(1)=i$, then try to prove $\sigma=(12\cdots n)^i$.