A study of enhanced power graphs of finite groups

Abstract

© 2020 World Scientific Publishing Company. The enhanced power graph e(G) of a group G is the graph with vertex set G such that two vertices x and y are adjacent if they are contained in the same cyclic subgroup. We prove that finite groups with isomorphic enhanced power graphs have isomorphic directed power graphs. We show that any isomorphism between undirected power graph of finite groups is an isomorphism between enhanced power graphs of these groups, and we find all finite groups G for which Aut(e(G) is abelian, all finite groups G with |Aut(e(G)| being prime power, and all finite groups G with |Aut(e(G)| being square-free. Also, we describe enhanced power graphs of finite abelian groups. Finally, we give a characterization of finite nilpotent groups whose enhanced power graphs are perfect, and we present a sufficient condition for a finite group to have weakly perfect enhanced power graph.