Automorphisms of valued Hahn groups

TL;DR

This paper investigates the automorphism groups of Hahn groups with canonical valuation, providing a structure theorem, characterizations, and matrix descriptions under certain conditions, advancing the understanding of their symmetries.

Contribution

It introduces a decomposition of automorphism groups of Hahn groups satisfying a lifting property, including characterizations and explicit matrix descriptions for special cases.

Findings

Automorphism group decomposes into a semidirect product under certain conditions.

Characterization of Hahn groups satisfying the lifting property.

Matrix descriptions of automorphism groups in special cases.

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.