Magnus varieties in group representations

L. E. Krop ; B. I. Plotkin

Sbornik. Mathematics, Tome 24 (1974) no. 4, pp. 487-510

Résumé

We consider varieties of pairs $(A,\Gamma)$, where $A$ is an Abelian group and $\Gamma$ is a group acting in $A$ as a group of automorphisms. In the semigroup of all such varieties we distinguish certain subsemigroups. If $\Theta$ is a group variety, we denote by $\omega'\Theta=\mathfrak X$ the variety of pairs $(A,\Gamma)$ such that if $(A,\overline\Gamma)$ is the corresponding faithful pair, then its corresponding semidirect product $A\leftthreetimes\overline\Gamma$ belongs to $\Theta$. We obtain a number of results concerning the operator $\omega'$. A pair $(A,\Gamma)$ 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 $\mathfrak X$ of pairs is a Magnus variety if all of its free pairs are Magnus pairs. We prove that if $\Theta$ is a polynilpotent group variety, then $\omega'\Theta$ is a Magnus variety. Bibliography: 17 titles.