Does the permutation group $S_8$ contain elements of order $14$?
My answer: If $\sigma =\alpha \beta $where $\alpha$ and $ \beta$ are disjoint cycles, then $|\sigma| =lcm(|\alpha|, |\beta|)$. Therefore the only possible disjoint cycle decompositions for a permutation $\sigma \in S_8$ with $|\sigma| =14$ is $(7,2)$. Since $7+2\neq 8$ so there is no element of order 14 in $S_8$.
Is my answer right? If no, what's the right answer?