Permutation help Consider the elements of $S_7$. For each $\sigma \in S_7$ there is a smallest positive integer |$\sigma$| such that $\sigma^{|\sigma|}=e$. Find the value of $N$= max{ $|\sigma|$ | $\sigma \in S_n$}. Give a specific permutation $\alpha \in S_7$ such that $|\alpha|=N.$

So honestly I have no idea where to even start. Any and all guidance is greatly appreciated.
 A: Your question is equivalent to find nonnegative integers $a_1+a_2...+a_k=7$ such that $lcm(a_1,,,a_k)$ is maximum.
To make it maximum,you should choose $a_i$ relativly prime as much as you can.After some try,you can see that answer is $12$.($a_1=3$ and $a_2=4$)
To generalize the result for $S_n$ may be challinging.
A: As mesel said, this is finding same as maximizing $lcm(a_1,..,a_k)$ for $a_i$ such that $a_1+a_2+...+a_k=7$. This follows from the fact that disjoint cycles commute and the order is then just the number of times a cycle must be composed with itself to make each of these disjoint cycles $e$.
But this is a hard question for general $n$- you know that once you have professional mathematicians working to find asymptotics it's a difficult problem.
But for $S_7$, we can see that the highest order is $12$ just by checking the conjugacy classes of n-cycles and fixed-points and finding the largest relatively prime integers. So the answer is $3 \times 4=12$ and we have $(1234)(567)$ as our $\sigma$. 
To build some intuition/number sense behind this, we also see that $3$ and $4$ are the integers that are closest to equal and still fit the conditions of adding up to $7$ and being coprime, so their product must be greatest of combinations $a_i$.
