Abstract

We examine the effect of the P-localization functor on three types of group extensions: extensions that give rise to a nilpotent action on the kernel, extensions with a nilpotent kernel and a torsion quotient, and extensions with a finite kernel.

Introduction

Assume P is a family of primes. A group G is said to be P-local if the function x ↦ xq from G to G is bijective for any prime q in the complement of P. As shown in, any group G can be mapped canonically into a unique P-local group GP; moreover, the assignment G ↦ GP defines a functor from the category of groups to the category of P-local groups. This functor is called the P-localization functor, and it plays an important role in homotopy theory (see and). For nilpotent groups the properties of the P-localization functor are well understood (see, and); however, its properties outside this subcategory remain largely a mystery.

One avenue to a more complete understanding of this functor is to determine its effect on short exact sequences. This is the aim of three papers, one by C. Casacuberta and M. Castellet and two by the author. The first of these examines the effect of P-localization on group extensions with a nilpotent kernel and a torsion quotient, the second looks at extensions with a finite kernel, and the third investigates extensions that give rise to a nilpotent action on the kernel.