ABSTRACT: Suppose G is not a nilpotent group of class at most n (a non nil-n group). Consider a subgroup H of G. In this paper, we introduce the relative non nil-n graph (n) H;G of a finite group G. It is a graph with vertex set GnC(n) G(H) and two distinct vertices x and y are adjacent if at least one of them belongs toH and [x; y] = 2 Zn 1(G), where the subgroup C(n) G(H) contains g 2 G such that [g; h] 2 Zn 1(G) for all h 2 H. We present some general information about the graph. Moreover, we define the probability which shows how close a group is to being a nil-n group. It is proved that two n-isoclinic groups which are not nil-n groups have isomorphic graphs under special conditions.

Keywords