Algebra group implies power degree group for field size

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., algebra group) must also satisfy the second group property (i.e., power degree group for a prime power)
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Statement

Suppose $N$ is a nilpotent associative algebra over a finite field $\mathbb{F}_q$ for a prime power $q$, and $G$ is the algebra group corresponding to $N$. Then, $G$ is a q-power degree group: all its degrees of irreducible representations are powers of $q$.