On the spectrum of sums of generators in a finite group

Let $K$ be any finite extension field of the field of rationals, and let $n$ and $\alpha_1,\dots,\alpha_n$ be given natural numbers. It is shown that there are only finitely many isomorphism classes of finite groups $G$ on $n$ generators $a_1,\dots,a_n$ such that the spectrum of the element $\sum\limits_{i=1}^n\alpha_ia_i$ of the algebra $\mathbf CG$ lies in $K$.