REAL CONNECTIVE K-THEORY AND THE QUATERNION GROUP

Let ko be the real connective K-theory spectrum. We compute koBG and koBG for groups G whose Sylow 2-subgroup is quaternion of order 8. Us ing this we compute the coefficients t(ko)(G) of the G fixed points o f the Tate spectrum t(ko) for G = Sl(2)(3) and G = Q(8). The results p rovide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that t(ko)G is not a w edge of Eilenberg-Mac Lane spectra, as occurs for groups of prime orde r.