Finite groups over arithmetical rings and globally irreducible representations

Given the ring of integers A of an algebraic number field K, for which natu ral number n is there a finite group G subset of GL(n, R) such that RG, the A-span of G, coincides with M(n, R), the ring of(n x n)-matrices over R? G iven G subset of GL(n, R) we show that RG = M(n, R) if and only if the Brau er reduction of G module every prime is absolutely irreducible. In addition , the question above is fully answered if n is an odd prime. (C) 1999 Acade mic Press.