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arXiv · 2401.05937

On the lattice of closed subgroups of a profinite group

Francesco de Giovanni·Iker de las Heras·Marco Trombetti
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Abstract

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.

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Francesco de Giovanni, Iker de las Heras, Marco Trombetti. 2024-01-11. On the lattice of closed subgroups of a profinite group. https://arxiv.org/abs/2401.05937

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