A note on the power graphs of finite nilpotent groups

Abstract

The power graph P(G) of a group G is the graph with vertex set G and two distinct
vertices are adjacent if one is a power of the other. Two finite groups are said to be
conformal, if they contain the same number of elements of each order. Let Y be a
family of all non-isomorphic odd order finite nilpotent groups of class two or p-groups of class less than p. In this paper, we prove that the power graph of each group in Y is isomorphic to the power graph of an abelian group and two groups in Y have isomorphic power graphs if they are conformal. We have determined the number of maximal cyclic subgroups of a generalized extraspecial p-group (p odd) by determining the power graph of this group. We have also determine the power graph of a p-group of order p4 (p odd).