Group Laws [x, y −1] ≡ u(x, y) and Varietal Properties

Abstract

Let F = ⟨x, y⟩ be a free group. It is known that the commutator [x, y −1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ⊆ F such that every law of the form [x, y −1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y −1] ≡ u, where u ∈ [F′, ⟨x⟩].