Finite arithmetic subgroups of GL_n

Abstract

We discuss the following conjecture of Kitaoka: if a finite subgroup $G$ of $G{L}_n(O_K)$ is invariant under the action of $Gal(K/\mathbb{Q})$ then it is contained in $G{L}_n(K^{ab})$. Here $O_K$ is the ring of integers in a finite, Galois extension $K$ of $\mathbb{Q}$ and $K^{ab}$ is the maximal, abelian subextension of $K$. Our main result reduces this conjecture to a special case of elementary abelian $p-$groups $G$. Also, we construct some new examples which negatively answer a question of Kitaoka.