Finite Groups over Arithmetical Rings and Globally Irreducible Representations,

Given the ring of integersRof an algebraic number fieldK, for which natural numbernis there a finite groupG GL(n, R) such thatRG, theR-span ofG, coincides withM(n, R), the ring of (n × n)-matrices overR? GivenG GL(n, R) we show thatRG = M(n, R) if and only if the Brauer reduction ofGmodulo every prime is absolutely irreducible. In addition, the question above is fully answered ifnis an odd prime.