Surjectivity need not be preserved by group localizations (Q1612117)

The author gives an answer to the following question: Is it possible that all idempotent functors in the category of groups carry surjective homomorphisms into surjective homomorphisms? The main result of this paper is the following theorem: Let \(L\) be any idempotent functor on groups. Then \(L\) preserves surjectivity if and only if the class of \(L\)-local groups has the following closure property: every subgroup of an \(L\)-local group which is also a quotient of an \(L\)-local group is \(L\)-local. Moreover, the author gives two examples proving that if this condition is not satisfied then an idempotent functor does not preserve surjectivity.