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$P$-localizing group extensions with a finite kernel

Published online by Cambridge University Press:  05 September 2005

KARL LORENSEN
Affiliation:
Mathematics Department, Penn State Altoona, Altoona, PA 16601-3760, U.S.A. e-mail: [email protected]

Abstract

Assume $P$ is a family of primes, and let $()_P$ represent the $P$-localization functor. If $1\,{\to}\,N\stackrel{\iota}{\to} G\stackrel{\epsilon}{\to} Q\,{\to}\,1$ is an exact sequence of groups with $N$ finite, we prove that the sequence $N_P\stackrel{\iota_P}{\to} G_P\stackrel{\epsilon_P}{\to} Q_P\,{\to}\,1$ is exact. Moreover, we provide an explicit description of $\mbox{Ker}\ \iota_P$ 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.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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