The lattice of algebraic closure operators on an infinite subgroup lattice

Abstract

We say a lattice L is a subgroup lattice if there exists a group G such that where is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of algebraic closure operators which act on the subgroup lattice of an infinite group is itself a subgroup lattice if and only if the group is isomorphic to the Prüfer p-group.