Cohomology and profinite topologies for solvable groups of finite rank

Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to C p ^az. We show that if G is nilpotent, then the pro-p completion map $G\to \hat {G}p$ induces an isomorphism $H\ast (\hat {G}p,M)\to H\ast (G,M)$ for any discrete $\hat {G}p$-module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map $H\ast (\hat {N}p,M)\to H\ast (N,M)$ is an isomorphism for any discrete $\hat {N}p$-module M of finite p-power order. Moreover, if G lacks any C p ^az-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.