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Automorphisms of valued Hahn groups

Salma Kuhlmann, Michele Serra

Abstract

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group $G$. Under the assumption that $G$ satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.

Automorphisms of valued Hahn groups

Abstract

Hahn groups endowed with the canonical valuation play a fundamental role in the classification of valued abelian groups. In this paper we study the group of valuation (respectively order) preserving automorphisms of a Hahn group . Under the assumption that satisfies some lifting property, we prove a structure theorem decomposing the automorphism group into a semidirect product of two notable subgroups. We characterise a class of Hahn groups satisfying the aforementioned lifting property. For some special cases we provide a matrix description of the automorphism group.
Paper Structure (7 sections, 12 theorems, 29 equations)

This paper contains 7 sections, 12 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. The lifting property
  4. Lifting from the rank to $o\text{-}\mathop{\mathrm{Aut}}\nolimits G$
  5. Lifting from the skeleton to $v\text{-}\mathop{\mathrm{Aut}}\nolimits G$
  6. Rayner groups
  7. Automorphism groups as groups of matrices

Key Result

Lemma 2.11

Let us keep the notation of Definition def:val-pres-iso. A valuation preserving isomorphism $\sigma\colon G_1\rightarrow G_2$ of Hahn groups naturally induces an isomorphism of the skeletons $[\sigma_\Gamma;\{\sigma_\gamma:\gamma\in\Gamma_1\}] \colon S(G_1)\rightarrow S(G_2)$ as follows: In particular, a valuation preserving automorphism $\sigma\in v\text{-}\mathop{\mathrm{Aut}}\nolimits G$ induc

Theorems & Definitions (49)

  • Definition 2.1: Hahn group
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9: salma-monograph
  • Definition 2.10
  • Lemma 2.11: salma-monograph
  • ...and 39 more