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