some results on isologism of pairs of groups

Let ν be a variety of groups defined by a set v of laws. then the verbal subgroup and the marginal subgroup of a group g associated with the variety are denoted by v (g) and v∗(g), respectively. let (n, g) be a pair of groups in which n is a normal subgroup of g. in the paper, we study the lower and upper ν-marginal series of the pair (n, g) and prove some properties of isologism of pairs of groups.