### Abstract

Assume P is a family of primes, and let ()_{P} represent the P-localization functor. If 1 → N →^{L} G, → ^{∈} Q → 1 is an exact sequence of groups with N finite, we prove that the sequence N_{P}^{LP} G_{P} →^{∈P} Q_{P}→ 1 is exact. Moreover, we provide an explicit description of Ker_{iP} when Q belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all P-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-nilpotent groups.

Original language | English (US) |
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Pages (from-to) | 193-206 |

Number of pages | 14 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 139 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 139, no. 2, pp. 193-206. https://doi.org/10.1017/S0305004105008509

**P-localizing group extensions with a finite kernel.** / Lorensen, Karl.

Research output: Contribution to journal › Article

TY - JOUR

T1 - P-localizing group extensions with a finite kernel

AU - Lorensen, Karl

PY - 2005/9/1

Y1 - 2005/9/1

N2 - Assume P is a family of primes, and let ()P represent the P-localization functor. If 1 → N →L G, → ∈ Q → 1 is an exact sequence of groups with N finite, we prove that the sequence NPLP GP →∈P QP→ 1 is exact. Moreover, we provide an explicit description of KeriP when Q belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all P-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-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 an exact sequence of groups with N finite, we prove that the sequence NPLP GP →∈P QP→ 1 is exact. Moreover, we provide an explicit description of KeriP when Q belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all P-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-nilpotent groups.

UR - http://www.scopus.com/inward/record.url?scp=33644616358&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644616358&partnerID=8YFLogxK

U2 - 10.1017/S0305004105008509

DO - 10.1017/S0305004105008509

M3 - Article

AN - SCOPUS:33644616358

VL - 139

SP - 193

EP - 206

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -