Let (G, N) be a pair of groups. In this article, first we construct a relative central extension for the pair (G, N) such that special types of covering pair of (G, N) are homomorphic image of it. Second, we show that every perfect pair admits...

Abstract. The paper devotes to derive certain upper bounds for the Schur multiplier

of a pair of ¯nite groups (G;N). Also we give a precise bound for the order of the

commutator subgroup [N;G] of a pair of groups in terms of the orders...

The main purpose of this paper is to prove some properties of a pair of groups from point of view of capability, the notion which was first introduced by G.Ellis [1] in 1996....

A group G is called capable if it is the group of inner automorphisms

of some group E. Capable pairs are defined in terms of a relative

central extension. In this paper we introduce the precise center

for a pair of groups...

Abstract

J.L. Loday (1978) introduced the concept of relative central extension and G. Ellis (1998) gave the notion

of covering pair of a pair of groups (G,N). In this paper under some condition we show the existence

of covering...

In 1998, G. Ellis defined the Schur multiplier of a pair $(G,N)$ of groups and

mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite $p$-groups $(G...

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...

A pair of groups (G,N) is said to be nilpotent if it

has a central series. A detailed study of nilpotent pair of groups

has been carried out recently. Related to, some further studies is

done in this paper. Topics include...

In this paper, we find an upper bound for the exponent of the Schur multiplier of a pair (G,N) of finite p -groups, when N admits a complement in G . As a consequence, we show that the exponent of the Schur multiplier of a pair (G,N) divides...