Some Remarks on the Varieties of Pairs of Groups

Let 𝑉 be a variety of groups defined by a set 𝑉 of laws .Let (𝑁,𝐺) be a pair of groups in which 𝑁 is a normal subgroup of 𝐺. we define the lower and upper Vmarginal series of the pair (𝑁,𝐺) and prove some results on the varieties nilpotent pais of groups. Moreover, we extend some properties of the Baerinvariant and isologism of a pairs of groups.