A note on Engel groups and local nilpotence

Abstract

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class <& including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class <€ is both finite-of-exponent-*?(n)-bynilpotent-of-class< c(n) and nilpotent-of-class< c(«)-by-finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

Group Laws Implying Virtual Nilpotence

“…In fact [1] gives in essence a characterization of words w yielding the conclusion of our Theorem A, but in terms of the form of w itself. However the results of [1,2,3] depended ultimately on a lemma of Shalev [20 [5] Group laws implying virtual nilpotence 299 apart from the improvements claimed for the results of the present paper over those of [1,2,3], it is also justified by the need to provide arguments based on the corrected version of Shalev's lemma. Nevertheless the arguments we use, although mostly self-contained, are to a considerable extent adapted from those of Shalev [20], as well as [1,2,15].…”

“…Theorem A is stronger than related results (for example of Point [15]) in that the nilpotency class c figuring in the conclusion, as well as the exponent e, depend only on the length of w. It is this that allows the conclusion to be extended to the full class [3] Group laws implying virtual nilpotence 297 5f, as opposed to just finitely generated soluble-by-finite groups, and in Theorem B below, where an additional condition is imposed on w, to the very large class of 'locally graded' groups. As a consequence of Theorem A and a result of Groves [4], we infer the following…”

On Positive Law Problems in the Class of Locally Graded Groups

“…Proof It is shown by Kim and Rhemtulla [5], that every locally graded nEngel group G is locally nilpotent. Then by a result of Burns and Medvedev [10], G is contained in a variety N c B e ∩ B e N c , where c and e depend on n only. By [3] the nilpotent variety N c satisfies some binary positive law, say P c (x, y) = Q c (x, y).…”

On difficult problems and locally graded groups

“…It is shown by Kim and Rhemtulla [11], that every locally graded n-Engel group G is locally nilpotent. Then by a result of Burns and Medvedev [6], G is contained in the variety N c(n) B e(n) ∩ B e(n) N c(n) , where c(n) and e(n) depend on n only. By [16], the variety N c(n) B e(n) satisfies a positive law, so does G.2 By A we denote the variety of all abelian groups, and by A p -the variety all abelian groups of exponent p. If a group G satisfies a positive law, then the variety var(G), it generates, has a basis of positive laws [14].…”