پایای بئر و گروههای پوششی تعمیم یافته

this thesis basically deals with the well-known notion of the bear-invariant of groups, which is the generalization of the schur multiplier of groups. in chapter two, section 2.1, we present an explicit formula for the bear-invariant of a direct product of cyclic groups with respect to nc, c>1. also in section 2.2, we caculate the baer-invatiant of a nilpotent product of cyclic groups wuth respect to nc, c>1. in section 2.3 a property for the baer-invariant of nilpotent product of arbitary groups will be presented which is somehow a generalization of haebich theorem [10].now, in chapter three, we will present some classes of groups which do not have any generalized covering group with respect to the variety nc, c>2. also we will construct a generalized group for some classes of groups. these results give a more clear idea of the subject than beyland tappes example [3] and also given an idea that wiegolds result [36] and haebichs result [10] can not be generalized to the variety of nilpotent groups of class at most c>2. in 1904 i. schur [30] showed that if g is a finite group and gp is a sylow p-subgroup of g, then m (g)p, the sylow p-subgroup of the schur multiplier of g, is isomorphic to a subgroup of m (gp). in 1973 j. wiegold and m. r. jones [17] presented an intersting generalization of schur theorem [18,30]. in fact they showed that if h is a subgroup of a finite group g of index n, then m (g). the n-th powers of elements of m (g), is embedded in m (h). in chapter four sections 4.1 and 4.2 we will generalize the last two results to the variety nc, c>1, and to the centre by centre an outer commutator word variety respectively. in section 4.3 we have shown that if g is a finite nilpotent group then for c+1 is a prime number. also we have shown that if g is a finite abelian group then for natural number c. finally, in section 4.4 we present an important example to show that the results of section 4.1, 4.2 and 4.3 are in the best possible situations. in fact, we given a non-niplotent, solvable group for which the result of section 4.3 and schur theorem [18,30] are not held.