I have been working on this problem for few hours, and have gotten no where. Here it is:
If $G$ is a $p$-group, and $G/\Phi(G)$ has order at least $p^{2\alpha-1}$, then the number of elements of order $p$ is congruent to $-1\mod p^\alpha$.
The hint is to mimic the proof of theorem 4.9. But there's so much stuff in that proof which doesn't apply to this case, so I do not know what to leave out and what to leave in. I have been able to figure out how to solve this problem, if only I can show that there is an irreducible character $\chi$ where $C$ acts trivially. Here $C$ is the group of linear characters $\lambda$ with $\lambda^p=1$, and it acts on characters $\chi$ by sending them to the product $\lambda\chi$. I also know $C$ is isomorphic to $G/\Phi(G)$. So my question is: how to show there exists a $\chi$ with $C$ acting trivially?