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.