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

Asked Mar 14 '19 at 16:55

Active Mar 14 '19 at 17:03

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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.

edited Mar 14 '19 at 16:59

asked Mar 14 '19 at 16:55

vesii

1,979

I think it will be $30.$ – little o Mar 14 '19 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 '19 at 17:00
Can you find an element of order $30$ in $S_{10}$? – little o Mar 14 '19 at 17:02
1

http://oeis.org/A000793 – Arthur Mar 14 '19 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 '19 at 17:03

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

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