The lattice of algebraic closure operators on an infinite subgroup lattice. (English) Zbl 07409824

Summary: We say a lattice \(L\) is a subgroup lattice if there exists a group \(G\) such that \(\mathrm{Sub}(G)\cong L,\) where \(\mathrm{Sub}(G)\) 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.


06A15 Galois correspondences, closure operators (in relation to ordered sets)
20D30 Series and lattices of subgroups
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