Bounded-index nilpotent subgroups of rational linear groups with finitely many automorphism orbits

Abstract

Let \(G\leq\operatorname{GL}_n(\mathbb Q)\) have finitely many orbits under its full abstract automorphism group, and put \(d=\dim_{\mathbb Q}\operatorname{span}_{\mathbb Q}G\). We prove that \(G\) has a torsion-free normal subgroup \(U\) satisfying \[ \gamma_n(U)=1,\qquad [G:U]\leq(2n+1)^d\leq(2n+1)^{n^2}. \] Thus \(G\) is virtually nilpotent, answering Kourovka Problem 21.40 affirmatively. More generally, if the trace set \(T=\{\operatorname{tr}(g):g\in G\}\) is finite, the same subgroup satisfies \([G:U]\leq |T|^d\). The proof combines bounded-degree roots of algebraic numbers with the trace radical of the rational matrix algebra generated by \(G\). Restriction of scalars gives corresponding bounds over number fields.