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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.

In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some projective special linear groups, Internat. J. Algebra Comput., 2012] the authors possed the following problem: Problem. Is there a simple group which is k-fold OD-characterizable for k ≥ 3 ?In this paper as the main result we give positive answer to the above problem and we introduce two simple groups which are k-fold OD-characterizable such that k ≥ 6.Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same order and degree pattern, Science China Mathematics, 2012], the authors possed two conjectures as follows: Conjecture 1. All alternating groups A m with m = 10 are OD-characterizable. Conjecture 2. All symmetric groups S m , with m = 10, are n-fold OD-characterizable, where n ∈ {1, 3}.In this paper we find some alternating and some symmetric groups such that these conjectures are not true for them.

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Let G be a nonabelian group, A ⊆ G an abelian subgroup and n 2 an integer. We say that G has an n-abelian partition with respect to A, if there exists a partition of G into A and n disjoint commuting subsets A1, A2, . . . , An of G, such that |Ai| > 1 for each i = 1, 2, . . . , n. We first classify all nonabelian groups, up to isomorphism, which have an n-abelian partition for n = 2, 3. Then, we provide some formulas concerning the number of spanning trees of commuting graphs associated with certain finite groups. Finally, we point out some ways to finding the number of spanning trees of the commuting graphs of some specific groups.