On certain group theoretical properties generalizing commutativity

Starting from properties of the commutativity graph of a group, we deal with two group theoretical classes containing all abelian groups: the class of commutative-transitive groups and that of power-commutative groups. We aim at collecting the main known results concerning the structure of these groups. Moreover, we give the definition of the commutative-transitive kernel and of the power-commutative kernel of any group. Finally, we point out properties of such kernels for some classes of groups.