Define the action of $S_n$ on $\mathbb{R}^n$:
take any $x\in S_n$, consider the mapping $x: \mathbb{R}^n\to\mathbb{R}^n$, $e_1, e_2 ...e_n$ are the standard basis of $\mathbb{R}^n$, $x(e_k)=e_{x(k)}$, clearly $x$ is a linear transformation.
How to show that if $x\in S_n$ is a reflection to some vector then $x$ is a transposition?
I can only show that transpositions are reflections, how to show that there are no more?