We extend arbitrary group completions to the category of pairs (G, N ) where G is a group and N is a normal subgroup of G. Relative localizations are special cases. Our construction is a grouptheoretical analogue of fibrewise completion and fibrewise localization in homotopy theory, and generalizes earlier work of Hilton and others on relative localization at primes. We use our approach to find conditions under which factoring out group radicals preserves exactness. This has implications in the study of the effect of plus-constructions on homotopy fibre sequences.

Infra-nilmanifolds are compact K(G, 1)-manifolds with G a torsion-free, finitely generated, virtually nilpotent group. Motivated by previous results of various authors on p-completions of K(G, 1)-spaces with G a finite or a nilpotent group, we study the homotopy type of p-completions of infra-nilmanifolds, for any prime p. We prove that the p-completion of an infra-nilmanifold is a virtually nilpotent space which is either aspherical or has infinitely many nonzero homotopy groups. The same is true for plocalization. Moreover, we show by means of examples that rationalizations of infra-nilmanifolds may be elliptic or hyperbolic.

After revisiting the concept of the torsion subgroup of a group with respect to a set of prime numbers P (as introduced by Ribenboim), we show that, for all p in P , p-th roots are unique in a virtually nilpotent group if and only if P -roots are unique in both its Fitting subgroup and its Fitting quotient. This more general notion of torsion also turns out to be sufficient to understand completely the kernel of the P -localization homomorphism of a virtually nilpotent group. Using this result, we characterize the finitely generated virtually nilpotent groups such that, when dividing out the P -torsion subgroup, P -roots exist and are unique in the resulting quotient.

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