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On the lattice of closed subgroups of a profinite group

Francesco de Giovanni, Iker de las Heras, Marco Trombetti

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.

On the lattice of closed subgroups of a profinite group

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.
Paper Structure (6 sections, 33 theorems, 39 equations)

This paper contains 6 sections, 33 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Lattices in a profinite group
  3. Distributivity
  4. Decomposability
  5. Finite width
  6. Modularity

Key Result

Lemma 2.1

Let $G$ be a profinite group and let $X$ be a closed subgroup of $G$. Then $X$ is open if and only if it is contained in only finitely many closed subgroups of $G$.

Theorems & Definitions (33)

  • Lemma 2.1
  • Corollary 2.2
  • Theorem 2.6
  • Lemma 2.7
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Theorem 4.1
  • Corollary 4.2
  • ...and 23 more