Finite arithmetic subgroups of GL_n

Details Finite arithmetic subgroups of GL_n Finite arithmetic subgroups of GL_n

1998-03-23

arxiv.org/abs/math/9803170v1

Mathematics

Number Theory

Scientific paper

We discuss the following conjecture of Kitaoka: if a finite subgroup $G$ of $GL_{n}(O_{K})$ is invariant under the action of $Gal(K/\Bbb Q)$ then it is contained in $GL_{n}(K^{ab})$. Here $O_{K}$ is the ring of integers in a finite, Galois extension $K$ of $\Bbb Q$ 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.

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