In Vinberg's textbook on algebra you are asked in an exercise to prove that the order of any element in the symmetric group on n letters does not exceed e^(n/e), which the book tells you is approximately 1.44^n.
This exercise was stated immediately after showing how to calculate the order of an element by writing it as a product of disjoint cycles and taking the least common multiple of the cycle lengths.
Any hints on how to do this? Unfortunately I've been stuck for a while.
Say you have the disjoint cycle deccomposition $\tau_1\cdots\tau_m$, with $$ \sum |\tau_i| = n$$
And you want to maximize $lcm(|\tau_i|)$. This is always less than $$ \prod |\tau_i| $$
Can you maximize this product, with the constraint $\sum |\tau_i| = n$? Hint: this is like maximizing the area of a box given its perimeter.
To maximize the product, we need all $|\tau_i|$ equal, so we can take $|\tau_i|=\frac{n}{m}$. [Note this is not necessarily an integer, but it doesn't matter.]
Now you can finish by maximizing the expression you just got, over all $m$:
Our maximized product is $\left(\frac{n}{m}\right)^m$. Since $n$ is fixed, we can take the derivative of this with respect to $m$ to find the maximum of this expression is achieved when $m=n/e$. So our overall maximum is $e^{n/e}$.
