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$P$-localizing group extensions with a finite kernel
Published online by Cambridge University Press: 05 September 2005
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
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 139 , Issue 2 , September 2005 , pp. 193 - 206
- Copyright
- 2005 Cambridge Philosophical Society
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