# What is the largest possible order of a permutation in $S_{n}$? [duplicate]

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.

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