The lattice of closure operators on a subgroup lattice. (English) Zbl 1507.06007

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.


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


[1] Birkhoff, G., On the combination of subalgebra, Proc. Cambridge Philos. Soc., 29, 441-461, (1933)
[2] Birkhoff, G., On the combination of topologies, Fundam. Math., 29, 156-166, (1936) · JFM 62.0688.05
[3] Birkhoff, G., Lattice Theory, (1979), American Mathematical Society, Providence, R.I. · Zbl 0126.03801
[4] Dwinger, Ph., On the closure operators of a complete lattice, Nederl. Akad. Wetensch. Proc. Ser. A. 57, Indag. Math., 16, 560-563, (1954) · Zbl 0056.26204
[5] Giacobazzi, R.; Palamidessi, C.; Ranzato, F., Weak relative pseudo-complements of closure operators, Algebra Univ., 36, 3, 405-412, (1996) · Zbl 0901.06003
[6] Gorbunov, V. A., Algebraic Theory of Quasivarieties, (1998), Siberian School of Algebra and Logic, Consultants Bureaur, New York
[7] Kilpack, M.
[8] Kilpack, M.; Magidin, A.
[9] Kilpack, M.; Magidin, A.
[10] Ore, O., Combinations of closure relations, Ann. Math. Second Ser., 44, 514-533, (1943) · Zbl 0060.06203
[11] Rotman, J. J., An Introduction to the Theory of Groups, (1995) · Zbl 0810.20001
[12] Ward, M., The closure operators of a lattice, Ann. Math., 42, 191-196, (1942) · Zbl 0063.08179
[13] Whitman, P., Lattices, equivalence relations and subgroups, Bull. Am. Math. Soc., 52, 507-522, (1946) · Zbl 0060.06505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.