This is from an old comprehensive exam:
Prove that any group with $p^n$ elements has a nontrivial center and use this to prove that any group with $p^n$ elements has at least $p$ inequivalent one-dimensional complex representations. Here $p$ is a prime number.
The part about having a nontrivial center is easily done with the class equation, but I'm having trouble with the part about $p$ inequivalent one-dim representations. The number of one dimensional representations is given by the order of the abelianization of the group: $$G/[G,G]$$ If the group is order $p$ or $p^2$ I can show that this holds, given that groups of order $p$ are cyclic, and therefore abelian, so the commutator subgroup is trivial. I also know that groups of order $p^2$ are abelian, so the commutator subgroup is again trivial.
But besides these cases I'm struggling to see how having a nontrivial center will yield the result. Any help would be appreciated!