First I have to compute the number of subgroup of order $3$ and $5$ of $S_5$ (the symmetric group of five elements) and $A_5$ (the alternant group of five elements) and I know that Sylow's theorems can help me in doing this however I am not very sure on how to proceed in the actual computations.
Then I have to explicitly build the subgroups of order $3$ and here I really don't know how to do this so any help will be greatly appreciated.
Hint:
The elements of order $\;3\;$ are the $\;3,\,-$ cycles only...and the same is true for $\;5\;$ . How many of these are there in $\;S_5\;$ , and how many of these belong to the same subgroup?
Of course, you can also use Sylow's Theorems to help you calculate the number of subgroups in each case.