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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory