ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

Let K/Q be a finite Galois extension with maximal order O-K and Galois group Gamma. For finite Gamma-stable subgroups G subset of GL(n)(O-K) it is known [4], that they are generated by matrices with coefficients in O-Kab, 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 rami. cation 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 subset of GL(n)(O-K) which are not fixed elementwise by the commutator subgroup of Gal(K/R).