University of Isfahan International Journal of Group Theory 2251-7650 2 1 2013 03 01 On finite arithmetic groups 199 227 2865 10.22108/ijgt.2013.2865 EN Dmitry Malinin I.H.E.S. Journal Article 2012 12 27 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 $g\in G$ 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 $G\subset GL_n(O_E)$ 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. https://ijgt.ui.ac.ir/article_2865_15eedcf8206252b2004de346b12153d2.pdf