bounds for the index of the second center subgroup of a pair of finite groups

By a pair of groups, we mean a group g and a normal subgroup n. ##in the present work, we give an upper bound for |n/z2(g,n)| in which z2(g,n) denote the second center subgroup of a pair (g,n) of finite groups where n is a subgroup of g. ##as a consequence, we obtain an upper bound for k/z(h,k) where h ≅g/z(g,n) and k ≅ n/z(g,n), for a pair (g,n) of finite groups