خواصی چند در مورد واریته گروهها

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.