Local nilpotency in varieties of groups with operators

A theorem of a rather general nature is proved, which gives a positive solution to the restricted Burnside problem for a variety of groups with operators whose identities are obtained by 'operator diluting' (in some precise sense) ordinary identities defining a variety of groups for which this problem has a positive solution. Namely, let $\Omega$ be a finite group, $V$ a family of $\Omega$-operator identities, and $\overline{V}$ a family of (ordinary) group identities obtained from $V$ by replacing all operators by 1. Suppose that the associated Lie ring of a free group in the variety $\overline{\mathfrak{M}}$ defined by $\overline{V}$ satisfies a system of multilinear identities that defines a locally nilpotent variety of Lie rings with a function $f(d)$ bounding the nilpotency class of a $d$-generator Lie ring in this variety. It is proved that if, for a $d$-generator $\Omega$-group $G$, the semidirect product $G\leftthreetimes\Omega$ is nilpotent, then the nilpotency class of $G$ is at most $f(d\cdot(|\Omega|^{|\Omega|}-1)/(|\Omega|-1))$.
A strong condition that $G\leftthreetimes\Omega$ be nilpotent is automatically satisfied if both $G$ and $\Omega$ are finite $p$-groups. Instead of the condition on the identities of the associated Lie ring, an analogous condition on the identities $\overline{V}$ could be required, but such a condition would be stronger. An example at the end of the paper shows that the word multilinear in this condition is essential. It is not yet clear whether the condition that $\Omega$ be finite is essential, and whether one can choose a function from the conclusion to be independent of $|\Omega|$. Earlier, in [1], a similar theorem on nilpotency in varieties of groups with operators was proved by the author. The author's results on groups with splitting automorphisms of prime order $p$ (see [2], [3]) are prototypes for both papers on operator groups.