Symmetric group $S_4$ . How many elements are there of order 4 Let $S_4$ be the symmetric group on 4 letters. How many elements of $S_4$ Have order 4?
I am learning group theory all by myself, and couldn't find a way to solve this problem.
I am aware about the fact that any permutation can be written as a product of disjoint cycle, and the order of that permutation is equal to the LCM of the disjoint cycles.
I am basically having the problem to find the permutations 
Please help me out
 A: Think like that-what are the possible cycle structures of the permutations in $S_4$? There are only five possible options:


*

*Four cycles of length $1$

*One cycle of length $2$ and two cycles of length $1$

*Two cycles of length $2$

*One cycle of length $3$ and one cycle of length $1$

*One cycle of length $4$
As you know, the order of a permutation equals to the lcm of the lengths of its disjoint cycles. So it is easy to check only the elements of the fifth type (one cycle of length $4$) are elements of order $4$. So all you need to find is how many cycles of length $4$ there are. Well, if you want $\sigma$ to be a $4$-cycle in $S_4$ you have $3$ options to choose the value of $\sigma(1)$, then $2$ options to choose what will be $\sigma(\sigma(1))$, and that's it. Once you know what are $\sigma(1)$ and $\sigma(\sigma(1))$ you know the whole permutation because it is a $4$-cycle and $\sigma(\sigma(\sigma(1)))$ must be the remaining element of $\{1,2,3,4\}$. So the number of such permutations is $3\times 2=6$. 
