Realizable classes of quaternion extensions of degree 4l

Let k be a number field and O-k its ring of integers. Let Gamma be the generalized quaternion group of order 4l, where l is an odd prime number. Let M be a maximal O-k-order in k[Gamma] containing O-k[Gamma] and Cl(M) its class group. We denote by R(M) the subset of Cl(M) formed by the realizable classes in the sense of McCulloh. In this article we define two subsets of R(M) and show that they are subgroups of Cl(M) provided that 2 and l are unramified in k/Q. (C) 2000 Academic Press.