A Carnot group G is a connected, simply connected, nilpotent Lie group with strati ed Lie algebra. We study intrinsic Lipschitz graphs and intrinsic di erentiable graphs within Carnot groups. Both seem to be the natural analogues inside Carnot groups of the corresponding Euclidean notions. Here 'natural' is meant to stress that the intrinsic notions depend only on the structure of the algebra of G. We prove that one codimensional intrinsic Lipschitz graphs are sets with locally nite G-perimeter. From this a Rademacher's type theorem for one codimensional graphs in a general class of groups is proved.

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called "groups of type ⋆" throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type ⋆ is rectifiable in the intrinsic sense (De Giorgi's rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.