Local nilpotency in varieties of groups with operators. (English. Russian original) Zbl 0820.20033
Russ. Acad. Sci., Sb., Math. 78, No. 2, 379-396 (1994); translation from Mat. Sb. 184, No. 3, 137-160 (1993).
Let \(\Omega\) be a finite group of operators and \(\mathcal M\) a variety of \(\Omega\)-groups, defined by the \(\Omega\)-identities \(\{v_ \alpha\}\). Then \(\{\overline{v_ \alpha}\}\) denotes the family of (ordinary) group identities which is obtained from \(\{v_ \alpha\}\) by replacing all operators from \(\Omega\) by \(1\) and \(\overline{\mathcal M}\) denotes the variety of groups defined by \(\{\overline{v_ \alpha}\}\). Suppose that locally nilpotent groups in \(\overline{\mathcal M}\) constitute a subvariety and that the associated Lie ring of a free group of \(\overline{\mathcal M}\) satisfies a system of multilinear identities that define a locally nilpotent variety of Lie rings in which the nilpotency class of a \(d\)- generator ring is bounded by a function \(f(d)\). Then it is proved in this paper that if a group \(G \in {\mathcal M}\) is such that the semidirect product of \(G\) and \(\Omega\) is locally nilpotent, then \(G\) belongs to a locally nilpotent variety in which the nilpotency class of a \(d\)- generator group is bounded by the function \(f(d \cdot (| \Omega|^{| \Omega|} - 1)/(| \Omega| -1))\). Some applications of the theorem are given for the case in which \(\Omega\) is a finite \(p\)-group. An example shows that the condition that the Lie ring identities be multilinear is necessary.
Reviewer: S.Oates-Williams (St.Lucia)
MSC:
20E25 | Local properties of groups |
20F19 | Generalizations of solvable and nilpotent groups |
17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
20E05 | Free nonabelian groups |
20E10 | Quasivarieties and varieties of groups |
20F40 | Associated Lie structures for groups |
20D15 | Finite nilpotent groups, \(p\)-groups |
20D45 | Automorphisms of abstract finite groups |
17B01 | Identities, free Lie (super)algebras |