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Research articles

ScienceAsia 38 (2012): 201-206 |doi: 10.2306/scienceasia1513-1874.2012.38.201


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 GC(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 gG such that [g,h]∈Zn−1(G) for all hH. 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.

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Department of Mathematics and Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author, E-mail: erfanian@math.um.ac.ir

Received 19 Jan 2012, Accepted 23 Apr 2012