Finite arithmetic subgroups of GL(n)

We discuss the following conjecture of Kitaoka: If a finite subgroup, G of GL(n)(O-K) is invariant under the action of Gal(K/Q) then it is contained i n GL(n)(K-ab). Here O-K is the ring of integers in a finite Galois extensio n K of Q and K-ab is the maximal abelian subextension of K. Our main result reduces this conjecture to a special case of elementary abelian p-groups G . Also, we construct some new examples which negatively answer a question o f Y. Kitaoka. (C) 1999 Academic Press.