Groups Satisfying Semigroup Laws, and Nilpotent-by-Burnside Varieties

Abstract

We investigate the structure of groups satisfying a positi¨e law, that is, an identity of the form u '¨, where u and¨are positive words. The main question here is whether all such groups are nilpotent-by-finite exponent. We answer this question affirmatively for a large class C C of groups including soluble and residually finite groups, showing that moreover the nilpotency class and the finite exponent in question are bounded solely in terms of the length of the positive law. It follows, in particular, that if a variety of groups is locally nilpotent-by-finite, then it must in fact be contained in the product of a nilpotent variety by a locally finite variety of finite exponent. We deduce various other corollaries, for instance, that a torsionfree, residually finite, n-Engel group is nilpotent of class bounded in terms of n. We also consider incidentally a question of Bergman as to whether a positive law holding in a generating subsemigroup of a group must in fact be a law in the whole group, showing that it has an affirmative answer for soluble groups. ᮊ 1997 Academic Press 510

Positive laws in fixed points

“…In accordance with the theorem by Burns, Macedońska and Medvedev [2] there exist n-bounded numbers c and e such that for any a ∈ A # the centralizer C G (a) is an extension of a nilpotent group of class at most c by a group of exponent dividing e. By Lemma 2.8 we conclude that the exponent of the quotient G/F (G) is {e, q}-bounded. Let {p 1 , .…”

“…In contrast with this negative result, Burns, Macedońska and Medvedev answered the question in the affirmative for a large class of groups including all solvable and residually finite groups [2]. In particular they showed that there exist functions c(n) and e(n) of n only, such that any finite group satisfying a positive law of degree n is an extension of a nilpotent group of class at most c(n) by a group of exponents dividing e(n).…”

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].…”

“…From this assumption and the law (13) which our p-group G satisfies, it follows that G satisfies a law of the form (14) where p 3 is a product of commutators in x and y (from F (2) ) of weights > c. We now impose a further lower bound on p, assuming in addition that p > e 4 .…”

On Positive Law Problems in the Class of Locally Graded Groups

“…It follows now from results of Kim and Rhemtulla ([5] Theorem A and Lemma 3), that G must be locally -(polycyclic-by-finite). Since G is soluble-by-finite and satisfies a positive law of degree n, then by ( [9], Theorem B), G must be periodic extension of a nilpotent group. In particular G ∈ N c B e , where c and e depend on n only, which finishes the proof.…”