some structural properties of upper and lower central series of pairs of groups

In this paper, we first present some properties of lower and upper central series of pair of groups. then the notion of n-isoclinism for the classification of pairs of groups is introduced, and some of the structural properties of the created classes are proved. moreover some interesting theorems such as baer theorem, bioch theorem, hirsh theorem for pair of groups are generalized. finally, it is shown that each n-isoclinism family of pairs contains a quotient irreducible pair.