Groups with a positive law on commutators. (Q1764822)

The laws considered in this paper are of the form \(\alpha(x_1,\dots,x_n)=\beta(x_1,\dots,x_n)\) without use of inverses. Here, in particular, laws of this form for variables \([a,b][c,d]^r\), \(a,b,c,d\in G\), \(r\in\mathbb{N}\), are considered. It is shown that this set of elements suffices to show that there is such a law also for all elements of \(G'\), and so \(G'\) must be nilpotent by (locally finite and of finite exponent) provided that \(G\) is residually a finite \(p\)-group. In the proofs some Engel conditions are mentioned.