Berkovich mentioned the following result of Mann in his book on p-groups:


*The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1.*


Do you know any reference for this statement? Also, I would like to ask you if a similar assertion holds for conjugacy classes.



Characters of p-groups
Here is a more elementary argument than Geoff's. The number of linear characters of $P$ divides the order of $P$ and is hence a power $p^r$ of $p$. All the other characters 
have degrees $d\_i$ of the form $p^{a\_i}$. Hence
 $$ p^n = p^r +\sum\_i p^{2a\_i}. $$
I claim that if $p^n$ is written as a sum of powers of $p$, then the number $N$ 
of summands satisfies $N\equiv 1$ (mod $p-1$), and the proof follows. One way to prove the claim is to consider the smallest summand $p^k$. Considering the sum mod $p^k$ shows that the number $N\_k$ of summands equal to $p^k$ is a multiple of $p$. Hence we can replace them
with $N\_k/p$ summands equal to $p^{k+1}$ without affecting the number of summands mod $p-1$. Now continue this argument until reaching $p^n$.


