On the structure of finite groups with dominatable enhanced power graph

Abstract

Let [Formula: see text] be a group. The enhanced power graph of [Formula: see text] is a graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text]. 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 [Formula: see text]-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 [Formula: see text] is dominatable if and only if there is a prime number [Formula: see text] such that [Formula: see text] and the Sylow [Formula: see text]-subgroups of [Formula: see text] are isomorphic to either a cyclic group or a generalized quaternion group.

The cyclic graph (deleted enhanced power graph) of a direct product

“…We need a reformulation of a result in our previous paper (Theorem 1.4 of [8]). We note that a result similar to Theorem 1.4 of [8] has been proved by Cameron in Theorem 9.1 (b) of [7] and by Mahmoudifar and Babai in the Main Theorem of [14]. We note that [14] was submitted in 2018, two years before [8] was submitted, even though it appeared after it.…”

A survey on enhanced power graphs of finite groups

“…Recently, Mahmoudifar & Babai [63] gave another characterization for finite groups with dominatable enhanced power graphs. The above result implies that if G is a group with |Z(G)| = 1, then P e (G) is not dominatable.…”

“…Then P e (G) is not dominatable. It was noted in [63] that if G is a finite group such that all its proper subgroups are dominatable, then P e (G) is not necessarily dominatable. Also, if there exists a nontrivial normal subgroup N of G such that both N and the quotient group G/N are dominatable, then P e (G) is not necessarily dominatable.…”

“…([63, Main theorem]) Let G be a finite group. Then P e (G) is dominatable if and only if there exists a prime p such that p | |Z(G)| and every Sylow p-subgroup of G is isomorphic to either a cyclic group or a generalized quaternion group, where Z(G) is the center of G.…”

“…([63, Theorem 3.18]) Suppose that G = G 1 ×G 2 ו • •×G r with gcd(|G i |, |G j |) = 1for distinct indices i and j. Then P e (G) is dominatable if and only if there exists k ∈ {1, 2, .…”