1. does the symmetric group S7 contain an element of order 5? of order 10? of order 15?
2. what is the largest possible order of an element of S7?
could anyone help me, please ??
Use the fact that the order of a permutation is the least common multiple of the lengths of its cycles.
The 7 elements must be partitioned into one of the following ways :
7
1+6
2+5
3+4
1+1+5
etc.
It's not hard to see that the least common multiple is maximized when using the partition 3+4=7, yielding a maximum order of $\displaystyle 3 \times 4 = 12$. So the product of a three-cycle with a four-cycle has order 12 and that is maximum.