The lattice of algebraic closure operators on an infinite subgroup lattice

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 g...

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Bibliographic Details
Published inCommunications in algebra Vol. 49; no. 7; pp. 2906 - 2915
Main Authors Kilpack, Martha L. H., Magidin, Arturo
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 18.06.2021
Taylor & Francis Ltd
Subjects
Algebra
Closure operator
Group theory
Operators
Primary 06A15
Secondary 20D30
subgroup lattice
Subgroups
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Summary: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.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2021.1883641