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In this paper, we study the tensor commutativity degree of pair of finite groups. Erdos introduced the relative commutativity degree and studied its properties. Then, Mr. Niroomand introduced the tensor relative commutativity degree, calculated tensor relative degree for some groups, and studied its properties. Also, he explained its relation with relative commutativity degree. In this paper, we give a brief explanation of the tensor commutativity degree and could introduce relations for this tensor commutativity degree. We proved some relations, including the relation among tensor commutativity degree of subgroups of a group. In addition, we introduced some boundaries for groups having special properties.

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In this paper, we give connection between the order of the generalized Baer-invariant of a pair of finite groups and its factor groups, when ν is considered to be the specific variety. Moreover, we give a necessary and sufficient condition in which the generalized Baer-invariant of a pair of groups can be embedded into the generalized Baer-invariant of pair of its factor groups.

In this paper we introduce the concept of generalized Baer-invariant of a pair of groups with respect to two varieties ν and w of groups. We give some inequalities for the generalized Baer-invariant of a pair of finite groups, when ν is considered to be the Schur-Baer variety. Further, we present a sufficient condition under which the order of the generalized Baer-invariant of a pair of finite groups divides the order of the generalized Baer-invariant of a pair of their factor groups. Moreover, we prove that the generalized Baer-invariant of a pair of groups commutes with direct limit.

In this paper, we introduce nilpotent and solvable pair of polygroups. By a pair of polygroups (P, N), we mean a polygroup P with a normal sub-polygroup N of P. In addition, we obtain a necessary and su cient condition between nilpotency (solvability) of pair of polygroups and pair of groups. In particular, we extend the theory of groups to a pair of polygroups. Finally, we study the relationship between nilpotent and solvable pair of poly-groups.

Let (G; N) be a pair of groups. In this paper,  rstly, we construct a relative central extension for the pair (G; N) such that special types of the covering pair of (G; N) are the homomor-phic image of it. Secondly, we show that every perfect pair admits at least one covering pair. Finally, among extending some proper-ties of perfect groups to perfect pairs, we characterize the covering pairs of a perfect pair (G; N) under some extra assumptions.

Let G be a finite p-group and N be a normal subgroup of G with |N|=pn and |G/N|=pm. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair (G, N) of finite p- groups is bounded by p1/2n(2m+n-1) and hence it is equal to p1/2n(2m+n-1)-t for some non-negative integer t. Recently, the authors have characterized the structure of (G, N) when N has a complement in G and t£3. This paper is devoted to classification of pairs (G, N) when N has a normal complement in G and t=4, 5.