# Number of inequivalent one-dimensional representations of a group of order $p^n$

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!

• $p$-groups are solvable (which indeed follows from the fact that the center is nontrivial), and so we must have $[G,G]\ne G$. Thus $G/[G,G]$ has order at least $p$.
– Mark
Feb 19, 2021 at 16:06

Mark already indicated that $$|G:G'|\geq p$$ and that is the proof, but if you want to construct linear representations, you might throw in a bit more sophistication for example if you have a character table at your disposal. Let $$H \lt G$$ with $$|G:H|=p$$ (such a maximal subgroup exists). Look at $$1_H^G$$, the principal character of $$H$$ induced to $$G$$. By Frobenius' Reciprocity $$[1_H^G,1_G]_G=[1_H,1_H]_H=1$$, so $$1_H^G=1_G + \sum_{\chi \in Irr(G)-\{1_G\}} a_{\chi}\chi$$, with $$a_{\chi}$$ non-negative integers. Comparing degrees: $$1_H^G(1)=|G:H|=p=1 + \sum_{\chi \in Irr(G)-\{1_G\}} a_{\chi}\chi(1)$$. And we conclude that if $$a_{\chi} \neq 0$$ then $$\chi(1) \lt p-1$$. Since $$\chi(1) \mid |G|$$, we must have $$\chi(1)=1$$ for all $$\chi$$ and hence $$1_H^G=1_G + \chi_2 + \cdots \chi_p$$, with all $$\chi_i$$'s linear. And so here are your at least $$p$$ linear representations!