A Jordan–Chevalley decomposition beyond algebraic groups

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 sub-group 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 foreach 𝑝 prime