Cohomology and Profinite Topologies for Solvable Groups of Finite Rank

Abstract

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 Cp∞. 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 Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

‘Cohomology and Profinite Topologies For solvable Groups of Finite Rank’

“…In the paper [1], a solvable FAR-group is a solvable group with finite abelian section rank. Moreover, FS denotes the class of all groups G such that, for each natural number n, G has only finitely many subgroups of index n. Proposition 2.7 in the paper states that every solvable FAR-group is a member of the class FS; however, the argument provided applies only when the group is abelian.…”