### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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**Cohomology and profinite topologies for solvable groups of finite rank.** / Lorensen, Karl.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Cohomology and profinite topologies for solvable groups of finite rank

AU - Lorensen, Karl

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866022815&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866022815&partnerID=8YFLogxK

U2 - 10.1017/S0004972711003340

DO - 10.1017/S0004972711003340

M3 - Article

AN - SCOPUS:84866022815

VL - 86

SP - 254

EP - 265

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 2

ER -