Relative group completions

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 group-theoretical 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.