On finite galois stable subgroups of gl_n in some relative extensions of number fields

Let K/ℚ be a finite Galois extension with maximal order O_K and Galois group Γ. For finite Γ-stable subgroups G ⊂ GL_n O_K) it is known [4], that they are generated by matrices with coefficients in O_{K ab}, K ab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups G ⊂ GL_n (O_K) which are not fixed elementwise by the commutator subgroup of Gal (K/R).