On varieties of groups satisfying an Engel type identity

Let m, n be positive integers, v   a multilinear commutator word and w=vmw=vm. Denote by v(G)v(G) and w(G)w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n  -Engel and w(G)w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n  -Engel and v(G)v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result remains valid with m allowed to be an arbitrary positive integer. In this direction, we show that if m, n are positive integers, u a multilinear commutator word and v the product of 896 u-words, then the class of all groups G   in which the vmvm-values are n  -Engel and the verbal subgroup u(G)u(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem C).