on finite arithmetic groups

let $f$ be a finite extension of $bbb q$‎, ‎${bbb q}_p$ or a global‎ ‎field 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 natural‎ ‎operation 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.