On finite arithmetic groups

begin{abstract} Let $F$ be a finite extension of $Bbb Q$, ${Bbb Q}_p$ or a globalfield of positive characteristic, and let $E/F$ be a Galois extension.We study the realization fields of finite subgroups $G$ of $GL_n(E)$ stable under the naturaloperation of the Galois group of $E/F$. Though for sufficiently large $n$ and a fixed algebraic number field $F$ every its finite extension $E$ is realizable via adjoining to $F$ the entries of all matrices $gin G$ for some finite Galois stable subgroup $G$ of $GL_n(Bbb C)$, there is only a finite number of possible realization field extensions of $F$ if $Gsubset GL_n(O_E)$ over the ring $O_E$ of integers of $E$. After an exposition of earlier results we give their refinements for the realization fields $E/F$. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.