On the Nilpotent Graph of a finite Group

Abstract: If G is a non-nilpotent group and nil(G) = {g \in G : <g, h> is nilpotent for all h\in G}, the nilpotent graph of G is the graph with set of vertices G-nil(G) in which two distinct vertices are related if they generate a nilpotent subgroup of G. Several properties of the nilpotent graph associated with a finite non-nilpotent group G are studied in this work. Lower bounds for the clique number and the number of connected components of the nilpotent graph of G are presented in terms of the size of its Fitting subgroup and the number of its strongly self-centralizing subgroups, respectively. It is proved the nilpotent graph of the symmetric group of degree n is disconnected if and only if n or n-1 is a prime number, and no finite non-nilpotent group has a self-complementary nilpotent graph. For the dihedral group Dn, it is determined the number of connected components of its nilpotent graph is one more than n when n is odd; or one more than the 2'-part of n when n is even. In addition, a formula for the number of connected components of the nilpotent graph of PSL(2,q), where q is a prime power, is provided. Finally, necessary and sufficient conditions for specific subsets of a group, containing connected components of its nilpotent graph, to contain one of its Sylow p-subgroups are studied; and it is shown the nilpotent graph of a finite non-nilpotent group G with nil(G) of even order is non-Eulerian.