MAGNUS VARIETIES IN GROUP REPRESENTATIONS

We consider varieties of pairs , where is an abelian group and is a group acting in as a group of automorphisms. In the semigroup of all such varieties we distinguish certain subsemigroups. If is a group variety, we denote by the variety of pairs such that if is the corresponding faithful pair, then its corresponding semidirect product belongs to . We obtain a number of results concerning the operator . A pair is called a Magnus pair if its lower stable series reaches zero at the first limit place and all factors of this series are free abelian groups. A variety of pairs is a Magnus variety if all of its free pairs are Magnus pairs. We prove that if is a polynilpotent group variety, then is a Magnus variety.Bibliography: 17 items.