Question

1.   Prove that a cycle of length l is odd if l is even.

Answer 1

Let $(a_1a_2...a_l)$ be a cycle of even length $l$. Taking into account that $(a_1a_2...a_l)=(a_1a_l)(a_1a_{l-1})...(a_1a_3)(a_1a_2)$, we conclude that this cycle is represented as a product of $l-1$ transpositions. Since $l$ is even, $l-1$ is odd, and thus $(a_1a_2...a_l)$ is odd.