  We introduce a notion of rectifiability modeled on Carnot groups. Precisely,
we say that a subset E of a Carnot group M and N is a subgroup of M, we say E
is N-rectifiable if it is the Lipschitz image of a positive measure subset of
N. First, we discuss the implications of N-rectifiability, where N is a Carnot
group (not merely a subgroup of a Carnot group), which include
N-approximability and the existence of approximate tangent cones isometric to N
almost everywhere in E. Second, we prove that, under a stronger condition
concerning the existence of approximate tangent cones isomorphic to N almost
everywhere in a set E, that E is N-rectifiable. Third, we investigate the
rectifiability properties of level sets of C^1_N functions, where N is a Carnot
group. We show that for almost every real number t and almost every
noncharacteristic point x in a level set of f, there exists a subgroup T_x of H
and r >0 so that f^{-1}(t) intersected with B_H(x,r) is T_x-approximable at x
and an approximate tangent cone isomorphic to T_x at x.
