Group localization and two problems of Levine

A. K. Bousfield’s H Z -localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine’s algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield H Z -localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group G to its Bousfield H Z -localization is not always a G -perfect subgroup. In the case of algebraic closure of groups, we prove the analogous result that this kernel is not always a union of invisible subgroups.