0
$\begingroup$

I was trying to solve the following question (from this link):

What is the largest possible order of a permutation in $S_{10}$?

I though there is some sophisticated way to solve it but then I read the solution (also located in the link). Is there a better, more sophisticated way to solve such problems for $S_n$, then writing down all the elements and their orders?

For example, finding the largest possible order of a permutation in $S_{101}$ will be much harder.

$\endgroup$
  • $\begingroup$ I think it will be $30.$ $\endgroup$ – Dbchatto67 Mar 14 at 16:59
  • $\begingroup$ @Dbchatto67 Hi, thanks for the reply. You are correct but how to calculate without writing down all the elements? $\endgroup$ – vesii Mar 14 at 17:00
  • $\begingroup$ Can you find an element of order $30$ in $S_{10}$? $\endgroup$ – Dbchatto67 Mar 14 at 17:02
  • 1
    $\begingroup$ oeis.org/A000793 $\endgroup$ – Arthur Mar 14 at 17:02
  • $\begingroup$ @Dbchatto67 Yes. But the question is how to find the largest order in $S_n$ (like $S_100$) without writing down all the ways to calculate $lcm$. $\endgroup$ – vesii Mar 14 at 17:03