A Jordan-Chevalley decomposition beyond algebraic groups

TL;DR

This paper extends the Jordan-Chevalley decomposition to definable groups in o-minimal structures, providing new structural insights and decompositions for classical groups beyond algebraic settings.

Contribution

It introduces a generalized Jordan-Chevalley decomposition for definable groups in o-minimal structures, including Levi and key decompositions for disconnected groups.

Findings

Definable linear groups are semidirect products of torsion-free and semialgebraic parts.

Established Levi and key decompositions in o-minimal structures.

Characterized definable p-groups and 0-groups, showing solvability and generation properties.

Abstract

We prove a decomposition of definable groups in o-minimal structures generalizing the Jordan-Chevalley decomposition of linear algebraic groups. It follows that any definable linear group G is a semidirect product of its maximal normal definable torsion-free subgroup N(G) and a definable subgroup P, unique up to conjugacy, definably isomorphic to a semialgebraic group. Along the way, we establish two other fundamental decompositions of classical groups in arbitrary o-minimal structures: 1) a Levi decomposition and 2) a key decomposition of disconnected groups, relying on a generalization of Frattini's argument to the o-minimal setting. In o-minimal structures, together with p-groups, 0-groups play a crucial role. We give a characterization of both classes and show that definable p-groups are solvable, like finite p-groups, but they are not necessarily nilpotent. Furthermore, we prove that definable p-groups (p=0 or p prime) are definably generated by torsion elements and, in definably connected groups, 0-Sylow subgroups coincide with p-Sylow subgroups for each p prime.