This question was previously asked at Math.SE, but didn't receive much attention.

Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, say $G \leq \operatorname{GL}_n(K)$ for some field $K$.

Question: Are there infinitely many primes $p$ such that $p$ divides $[G:N]$ for some normal subgroup $N < G$ of finite index?

This seems to be true when $\operatorname{char} K = 0$, by 10.4 in "Infinite Linear Groups" by Wehrfritz.

In case the formulation of the question is unclear, here is a rephrasing. Let $\mathscr{N}$ be the set of all normal subgroups of $G$ with finite index. Is the following set of primes infinite? $$\{p \in \mathbb{Z}: p \text{ is a prime, and } p \mid [G:N] \text{ for some } N \in \mathscr{N} \}$$