Prove that cycle $(a_1a_2...a_k)$ is even $\iff$ $k$ is odd.
This makes intuitive sense because $(a_1a_2a_3...a_k)=(a_1a_k)(a_1a_{k-1})...(a_1a_3)(a_1a_2)$ which will be an even number of transpositions when $k$ is odd, and odd number when $k$ is even. I am hoping for something slightly more rigorous, perhaps using the definition of an orbit of $S_n$.