A generalization of group commutativity

In this paper, we give a generalization of the concept of group commutativity. A group [Formula: see text] will be defined as [Formula: see text]-abelian (or [Formula: see text]-commutative), where [Formula: see text] is the alternating group of degree [Formula: see text], if [Formula: see text], for every [Formula: see text], and every pairwise non-commuting elements [Formula: see text]. We prove that [Formula: see text]-commutativity implies [Formula: see text]-commutativity for every [Formula: see text]. For [Formula: see text]-groups, we prove that the converse holds. We provide some sufficient condition for [Formula: see text]-commutativity, and then we prove that Hamiltonian groups and minimal non-abelian [Formula: see text]-groups are [Formula: see text]-abelian. For finite groups, we prove that a non-abelian and [Formula: see text]-abelian group is nilpotent of class two and has even order. A partial description of such groups is given.