Frattini Extensions and Class Field Theory

A. Brumer has shown that every profinite group of strict cohomological p-dimension 2 possesses a class field theory the tautological class field theory. In particular, this result also applies to the universal p-Frattini extension G̃p of a finite group G. We use this fact in order to establish a class field theory for every p-Frattini extension π : G̃ → G (Thm.A). The role of the class field module will be played by the p-Frattini module. The universal norms of this class field theory will carry important information about the p-Frattini extension π : G̃ → G. A detailled analysis will lead to a characterization of finite groups G which have a p-Frattini extension π : G̃ → G in which G̃ is a weakly-orientable p-Poincaré duality group of dimension 2 (Thm.B). In section §5 we characterize the p-Frattini extensions πA1 : Sl2(Zp) → Sl2(Fp), p = 2, 3, 5, by some kind of localization technique. This answers a question posed by M.D.Fried and M.Jarden (Thm.C). It is quite likely that such an approach might also be successful for the characterization of the pFrattini extensions πD : XD(Zp) → X(Fp), where XD is the simple simplyconnected split Z-Chevalley group scheme with Dynkin diagram D.