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Jordan-Holder Theorem for profinite groups and applications

Tamar Bar-On, Nikolay Nikolov

TL;DR

This paper generalizes the Jordan-Holder theorem to profinite groups by introducing accessible and composition series and proving that the resulting composition factors form a well-defined multiset independent of the series. It then applies these results to infinite prosolvable Galois extensions, proving that a separable extension K/F is prosolvable if and only if it is solvable by radicals, via transfinite radical towers and adjoining all roots of unity. The work also analyzes the relationship between topological and abstract composition factors, showing that every abstract factor appears as a section of a topological factor and giving partial results (notably in the anabelian setting) and highlighting open questions about their equivalence.

Abstract

We generalize the notions of composition series and composition factors for profinite groups, and prove a profinite version of the Jordan-Holder Theorem. We apply this to prove a Galois Theorem for infinite prosolvable extensions. In addition, we investigate the connection between the abstract and topological composition factors of a nonstrongly complete profinite group.

Jordan-Holder Theorem for profinite groups and applications

TL;DR

This paper generalizes the Jordan-Holder theorem to profinite groups by introducing accessible and composition series and proving that the resulting composition factors form a well-defined multiset independent of the series. It then applies these results to infinite prosolvable Galois extensions, proving that a separable extension K/F is prosolvable if and only if it is solvable by radicals, via transfinite radical towers and adjoining all roots of unity. The work also analyzes the relationship between topological and abstract composition factors, showing that every abstract factor appears as a section of a topological factor and giving partial results (notably in the anabelian setting) and highlighting open questions about their equivalence.

Abstract

We generalize the notions of composition series and composition factors for profinite groups, and prove a profinite version of the Jordan-Holder Theorem. We apply this to prove a Galois Theorem for infinite prosolvable extensions. In addition, we investigate the connection between the abstract and topological composition factors of a nonstrongly complete profinite group.

Paper Structure

This paper contains 3 sections, 16 theorems, 10 equations.

Table of Contents

  1. Composition factors and Jordan-Holder Theorem
  2. Galois Theorem for infinite extensions
  3. Topological vs. Abstract Composition Factors

Key Result

Theorem 1

Let $G$ be a finite group and let $\{e\}=G_n \unlhd \cdots G_1\unlhd G_0=G$ and $\{e\}=H_m \unlhd \cdots H_1\unlhd H_0=G$ be two composition series for $G$. Then for every finite simple group $S$, $|\{0\leq i\leq n-1: G_i/G_{i+1}\cong S\}|=|\{0\leq i\leq m-1: H_i/H_{i+1}\cong S\}|$. In particular $m

Theorems & Definitions (37)

  • Theorem 1: Jordan-Holder Theorem for finite groups
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Corollary 5
  • Theorem 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 27 more