## 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

### Keywords:

closure operators; lattice; subgroups
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### References:

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