# Possible orders of permutations of 11 symbols

Which of the following numbers can be orders of permutations \alpha of 11 symbols such that it does not fix any symbol?
1. 18,
2. 30,
3. 15,
4. 28

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A permutation $\pi$ can always be written as product of disjoint cycles $$\pi=c_1\cdot\ldots\cdot c_k$$ and then the order of $\pi$ is the lcm of the lengths of the cycles $c_1,\dots,c_k$.

Thus you are looking for partitions $$11=\ell_1+\dots+\ell_k$$ with $\ell_i\geq2$ for all $i$ such that $lcm(\ell_1,...,\ell_k)$ is some specified value.

For instance $11=5+6$ proves that there's a permutation as required of order 30.

Try to work out the other values by yourself!

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yes 18,24,28,30 are the possible orders ,thank you very much. –  antara Dec 19 '12 at 14:09
what about $15$??????? –  La Belle Noiseuse May 31 '13 at 18:03
@TaxiDriver : $11=5+3+3$ and $lcm(5,3,3)=15$. –  Andrea Mori Jun 1 '13 at 9:20
thankkuuuuuuuuuuuu –  La Belle Noiseuse Jun 1 '13 at 9:24