### Abstract

Assume P is a family of primes, and let ()_{P} represent the P-localization functor. If 1 → N → ^{l} G → ^{∈} Q → 1 is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence N_{P} → ^{lP} G_{P} → ^{∈P} Q_{P} → 1 is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ι_{P} is an injection. This generalizes the well-known result that ()_{P} is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups.

Original language | English (US) |
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Pages (from-to) | 4345-4364 |

Number of pages | 20 |

Journal | Communications in Algebra |

Volume | 34 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2006 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Communications in Algebra*, vol. 34, no. 12, pp. 4345-4364. https://doi.org/10.1080/00927870600936609

**P-localizing group extensions with a nilpotent action on the kernel.** / Lorensen, Karl.

Research output: Contribution to journal › Article

TY - JOUR

T1 - P-localizing group extensions with a nilpotent action on the kernel

AU - Lorensen, Karl

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Assume P is a family of primes, and let ()P represent the P-localization functor. If 1 → N → l G → ∈ Q → 1 is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence NP → lP GP → ∈P QP → 1 is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ιP is an injection. This generalizes the well-known result that ()P is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups.

AB - Assume P is a family of primes, and let ()P represent the P-localization functor. If 1 → N → l G → ∈ Q → 1 is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence NP → lP GP → ∈P QP → 1 is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ιP is an injection. This generalizes the well-known result that ()P is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups.

UR - http://www.scopus.com/inward/record.url?scp=33845725539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33845725539&partnerID=8YFLogxK

U2 - 10.1080/00927870600936609

DO - 10.1080/00927870600936609

M3 - Article

AN - SCOPUS:33845725539

VL - 34

SP - 4345

EP - 4364

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 12

ER -