Cohomology ring of the generalized quaternion group with coefficients in an order

We determine the ring structure of the cohomology H*(Q(2r), (psi) Gamma) of the generalized quaternion group Q(2r) of order 2(r+2) by means of a diagonal approximation on the periodic resolution of period 4, where the, coefficient module (psi)Gamma is an order of a simple component of the group ring QQ(2r). Simultaneously, we give the precise description of the integral cohomology ring H*(Q(t), Z) for arbitrary generalized quaternion groups Q(t) of order 4t for any positive integer tgreater than or equal to2.