Some Structural Properties of Upper and Lower Central Series of Pairs of Groups

Abstract

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.

Keywords

• ‎n-isoclinism
• pair of groups
• quotient irreducible pair
• pi-groups