### 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 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.

Original language | English (US) |
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Pages (from-to) | 254-265 |

Number of pages | 12 |

Journal | Bulletin of the Australian Mathematical Society |

Volume | 86 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1 2012 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)