  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.
