What is the largest subgroup of $\displaystyle S_{5}$ that is also a primary group. (prime group) and how do you find it. What about for $\displaystyle S_{6}$.
I am not sure what you mean by "primary". If you mean a subgroup of prime order then that is easy to answer. Such as subgroup must be generated by an element of order 5 i.e. a 5-cycle. Thus, $\displaystyle (12345)$ is an element of order 5 and so $\displaystyle \left< (12345)\right>$ is a primary subgroup.
Since $\displaystyle |S_5|=5! = 2^3\cdot 3\cdot 5$ so we can look for Sylow $\displaystyle 2,3,5$ subgroups.
For the Sylow subgroups of $\displaystyle S_5$ it has order $\displaystyle 5$ and so is cyclic. Therefore, the Sylow 5-subgroups are generated by a 5-cycle. The same with $\displaystyle p=3$. With $\displaystyle p=2$ its gets a lot more involved, you can get many dihedral subgroups for example.