Abstract: Firstly, by introducing the notion of distributors of groups, we give some properties of distributors of groups. Secondly, we generalize  the corresponding results of commutators and p-commutators in group theory, and  prove that the mapping f from group G to group H is a group homomorphism if and only if the f-distributor of group G is 1. Finally, as an application, we calculate the number of homomorphisms from a class of metacyclic groups to dihedral groups.

Key words: distributor, p-commutator, p-derived group, group homomorphism