Let $G$ be an infinite finitely generated group and suppose that $G$ is linear, i.e., that $G \leq \operatorname{GL}_n(K)$ for some field $K$.
Are there infinitely many primes $p$ such that $p$ divides $[G:N]$ for some normal subgroup $N < G$ of finite index?
I think this is true at least if $\operatorname{char} K = 0$, by Theorem 10.4 in the book "Infinite Linear Groups" by Wehrfritz.
EDIT: To clarify, let me rephrase the question. Let $\mathscr{N}$ be the set of all normal subgroups of $G$ with finite index. Is the following set of primes infinite? $$\pi = \{p \in \mathbb{Z}: p \text{ is a prime, and } p \mid [G:N] \text{ for some } N \in \mathscr{N} \}$$