Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols Which of the following numbers can be orders of a permutation $\sigma$ of $11$ symbols, such that $\sigma$ does not fix any symbols?
$1. \;18$
$2.\; 30$
$3.\;15$
$4.\; 28$
could any one just give me hints?
 A: Think about the orders of the permutations (1 2) (3 4 5) and (1 2) (3 4 5 6) and see if you can think of a rule that gives the order of a permutation written in disjoint cycle notation. Now if $\sigma$ fixes no points, what does it tell you about the permutation written in disjoint cycle notation? From this you should be able to tell which orders are possible.
A: Firstly find all partitions of 11 as follows:
$$\begin{array}{ll}
                                 & \text{ corresponding l.c.m }   \\       
   11                          &11\\
10+1.                        &10 (\text{l.cm of }10,1)\\
9+2.                          & 18 \text{l.cm of }9,2 \\
9+1+1\\
8+3\\
8+2+1\\
8+1+1+1\\
.\\
.7+4\\
.\\
.\\
6+5\\
6+4+1\\
.\\
.\\
5+5+1\\
.....\\
5+ 3+2\\
5+3+1+1+1.    &          15( \text{l.c.m of }  5,3,1,1)\\
.......\\
\text{Similarly up to } 1+1.....+1 \ (11 \text{times})\\
\end{array}    $$
When we calculate l.c.m in all above cases we see 28 l.c.m is not possible 
So possible answers are 18,15,30
