Denote by $S\_n$ the group of permutations of the set $\{1,\ldots,n\}$ with composition as binary operation. Let $m\_n$ denote the maximum order that an element of $S\_n$ can have. What is the smallest positive integer $k$ such that $\lim\_{n\to\infty} \frac{m\_n}{n^k} < \infty$?



Maximum element order in $S_n$
Landau proved in 1902 that the maximal order of an element in $S\_n$ is $e^{(1+o(1))\sqrt{n\log n}}$. In particular, there is no integer $k$ with the property you ask for.


