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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]∉Z_n−1(G), where the subgroup C⁽ⁿ⁾_G(H) contains g∈G such that [g,h]∈Z_n−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.

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

Received 19 Jan 2012, Accepted 23 Apr 2012