Semigroup identities in 4-Engel groups

Abstract

We say that a group G or a variety of groups V satisfies a semigroup law, if it satisfies a nontrivial law of the form u(x1, . . . , xn) = v(x1, . . . xn), where u and v are words in the free semigroup freely generated by x1, . . . , xn. It follows from a result of J. Lewin and T. Lewin [2] that a variety V of groups which satisfies a semigroup law can be characterised by its semigroups laws. Furthermore, we have then a sufficient and necessary condition for a semigroup to be embeddable in some group in V . A semigroup S is embeddable in some group in V if and only if it is cancellative and it satisfies all the semigroup laws that hold in V . In other words we have that S is embeddable in some group in V if and only if S is a cancellative semigroup in the corresponding semigroup variety. In [4] B. H. Neumann and T. Taylor show that nilpotent groups satisfy semigroup laws. We will be using their work later on so we will now describe it in more details. Let F be a free group that is freely generated by the variables x, y, z1, z2, . . .. We define a sequence of words q1, q2, . . . in the variables x, y, z1, z2, . . . by induction as follows.