19.79 (2018)

(T. Springer). Let $G$ be a group. Suppose that for every integer $n > 0$ the group $G$ has a unique (up to isomorphism) irreducible complex linear representation of dimension $n$. (Note that $G = \text{SL}(2, \mathbb{Q})$ has these properties, by a theorem of Borel–Tits (Ann. Math., 97 (1973), 499–571).) Is it true that $G$ has a normal subgroup $N$ such that $G/N$ is isomorphic to $\text{SL}(2, \mathbb{Q})$?