Nilpotent groups, o-minimal Euler characteristic, and linear algebraic groups

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, the nilpotent case. It turns out that context, like for finite nilpotency is equivalent to normalizer property or uniqueness of Sylow subgroups, provided maximal normal torsion-free subgroup nilpotent. As consequence, we show decompositions prove Lie group an expansion reals if only it isomorphic group.