ON FINITE ARITHMETIC GROUPS

Let f be a finite extension of q, qp or a global field of positive characteristic, and lete=f be a galois extension. we study the realization fields of finite subgroups g of gln(e) stableunder the natural operation of the galois group of e=f. though for suffciently large n and a fixedalgebraic number field f every its finite extension e is realizable via adjoining to f the entries of allmatrices g 2 g for some nite galois stable subgroup g of gln(c), there is only a finite number ofpossible realization field extensions of f if g  gln(oe) over the ring oe of integers of e. after anexposition of earlier results we give their refinements for the realization fields e=f. we consider someapplications to quadratic lattices, arithmetic algebraic geometry and galois cohomology of relatedarithmetic groups.