On the Hyper Marginal Groups and Perfect Groups in an Arbitrary Variety of Groups

Let be a variety of groups defined by a set V of laws. In this study, for infinite groups, we first continue the hyper marginal series of infinite ordinal numbers via transfinite recursion and by this idea introduce the hyper marginal subgroup of an arbitrary group then with define the first hyper marginal groups, extend the Grun Lemma in an arbitrary varity of groups and show that all of perfect groups in variety (- perfect groups) are first hyper marginal groups. In the end of this investigation, we compute hyper marginal subgroup of the quotient group and direct product of a finite collection of perfect groups.