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The lattice of closure operators on a subgroup lattice. (English) Zbl 06891554

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 closure operators which act on the subgroup lattice of a finite group \(G\) is itself a subgroup lattice if and only if \(G\) is cyclic of prime power order.

MSC:

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