ScienceAsia 38(2012): 201-206 |doi:
Relative non nil-n graphs of finite groups
Ahmad Erfanian*, Behnaz Tolue
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 Γ(n)H,G of a finite group G. It is a graph with vertex set G∖C(n)G(H) and two distinct vertices x and y are adjacent if at least one of them belongs to H and [x,y]∉Zn−1(G), where the subgroup C(n)G(H) contains g∈G such that [g,h]∈Zn−1(G) for all h∈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.
|Department of Mathematics and Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran
* Corresponding author, E-mail: email@example.com
Received 19 Jan 2012, Accepted 23 Apr 2012