Let G be a finite group, let p be a prime and let w be a group-word. We say that G satisfies P (w, p) if the prime p divides the order of xy for every w-value x in G of p ′ -order and for every non-trivial w-value y in G of order divisible by p. With k ≥ 2, we prove that the kth term of the lower central series of G is p-nilpotent if and only if G satisfies P (γ k , p). In addition, if G is soluble, we show that the kth term of the derived series of G is p-nilpotent if and only if G satisfies P (δ k , p).Here o(x) denotes the order of the group element x. We point out that, instead of studying the couple of elements (x, y), where x is a p ′ -element of prime power order and y is a non-trivial p-element, one can focus on the couple (x, y), where x is a p ′ -element and y is a non-trivial element of order divisible by p. Corollary B.Let G be a finite group and let p be a prime. Then G is p-nilpotent if and only if for every x ∈ G such that p does not divide o(x) and for every 1 = y ∈ G such that p divides o(y), p divides o(xy).

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