Positive laws on large sets of generators: counterexamples for   infinitely generated groups
  Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with. 
C. Acciarri, G.A. Fern\'andez-Alcober
math.GR