Non-Nilpotent Graph of a Group

Abstract

We associate a graph N G with a group

Ensuring a Group is Weakly Nilpotent

“…Abdollahi and Zarrin in [4] studied the influence of G on the structure of G . Here, we obtain PSL 2 q , where PSL n q is the projective special linear group of degree n over the finite field of size q, which is a key result to the proof of Theorem 4.4, below.…”

“…Let G be a group. Following [2] and [4], we shall use the notation G ( G ) to denote the nonnilpotent graph (non-commuting graph, respectively) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent (non-commuting, respectively) subgroup.…”

The non-nilpotent graph of a semigroup

“…The former is excluded as a group of order p is commutative and thus it has an empty lower non-nilpotent graph (and thus not a complete graph). Hence, G = {e} and L S is complete by part (1).…”

“…In the past twenty years fascinating questions and results have been raised and investigated by studying graphs associated to groups or rings (see for example [1,2,5,6,7,9,19,23,24]). In [1] the notion of a non-nilpotent graph N G of a group G is introduced. The vertices of the graph are the elements of G and there is an edge between vertices if they do not generate a nilpotent group.…”

“…One of the results proved in [1] is that the number of connected components of N G for a finite group G is either Z * (G) or Z * (G) +1, where Z * (G) denotes the hypercenter of G. Note the elements of Z * (G) are isolated points in the graph N G . Hence, in [1], one also studied the induced subgraph N G on G nil(G), where nil(G) is the subset of those elements g ∈ G such that the group generated by g and h is nilpotent for any h ∈ H. In general it is unknown whether nil(G) is a subgroup of G, but in many important cases it is. For example, nil(G) = Z * (G) if G is a finite group (or more general, if G satisfies the maximal condition on its subgroups) or if G is a finitely generated solvable group.…”