What is the largest subgroup of that is also a primary group. (prime group) and how do you find it. What about for .
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, is an element of order 5 and so is a primary subgroup.
Since so we can look for Sylow subgroups.
For the Sylow subgroups of it has order and so is cyclic. Therefore, the Sylow 5-subgroups are generated by a 5-cycle. The same with . With its gets a lot more involved, you can get many dihedral subgroups for example.