  We show that strictly abnormal geodesics arise in graded nilpotent Lie
groups. We construct such a group, for which some Carnot geodesics are strictly
abnormal; in fact, they are not normal in any subgroup. In the step-2 case we
also prove that these geodesics are always smooth. Our main technique is based
on the equations for the normal and abnormal curves, that we derive (for any
Lie group) explicitly in terms of the structure constants.
