Groups Satisfying Semigroup Laws, and Nilpotent-by-Burnside Varieties Original Research Article

We investigate the structure of groups satisfying apositive law, that is, an identity of the formu ≡ v, whereuandvare 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 image 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 torsion-free, residually finite,n-Engel group is nilpotent of class bounded in terms ofn. 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.