On the structure of finite groups with dominatable enhanced power graph

چکیده مقاله

Let G be a group. The enhanced power graph of G is a graph with vertex set G and two distinct vertices x and y are adjacent if there exists z ϵ G such that x = zm and y = zn for some m,n ϵ ℕ. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite p-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group G is dominatable if and only if there is a prime number p such that p ||Z(G)| and the Sylow p-subgroups of G are isomorphic to either a cyclic group or a generalized quaternion group. © 2022 World Scientific Publishing Company.