chapter one is devotod to collect some notion and background informations, which are needed in the next chapters. it also contains some important statements which will be proved in a more general context later in this thesis. in chapter two, we show that if the marginal factor-group is of order np1...pk,n>1, then we obtain a bound for the order of the verbal subgroup. also a bound for the bear-invariant of a finite p-group with respect to the variety of polynilpotent groups of a given class row will be constructed. chapter three is devoted to present some inequalities for the bear-invariants of a finite group, with respect to a given variety of groups. using these results a genaralized version of a theorem of stallings (19965) will be proved. it is also given a sufficient condition for a family of v-nilpotent groups, which does not have any v-covering groups, with respect to a certain variety of gruops v. in chapter four, we study the concepts of v-islogisms and v-marginal extensions of groups, with respect to a given variety of groups v. finally we give equivalent conditions under which two extensions are v-isologic.