P-localization of relative groups

A relative group is a pair (N,E) consisting of a group E and a normal subgr oup N of E. Localization at sets of primes with respect to any radical is i ntroduced as a functor on the category of relative groups. Given a radical R and a set of primes P, we consider the full subcategory of relative group s (N,E) which are P-local with respect to R, i.e. N is a P-local group and R(EIN) is trivial. We construct a left adjoint to the inclusion of this sub category into the category of relative groups. As an application, when the radical R assigns the P'-isolator or the P'-torsion subgroup to a group, th e effect of the classical P-localization is made explicit for groups fittin g into an extension with a quotient F such that F/RF is torsion. (C) 2001 E lsevier Science B.V. All rights reserved.