In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

Baumslag and Wiegold have recently proven that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. Motivated by this surprisingly new result, we have obtained related results that just consider sets of prime divisors of element orders. For instance, the first of our main results asserts that G is nilpotent if and only if π(o(xy)) = π(o(x)o(y)) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold Theorem. While this result is still elementary, we also obtain local versions that, for instance, characterize the existence of a normal Sylow p-subgroup in terms of sets of prime divisors of element orders. These results are deeper and our proofs rely on results that depend on the classification of finite simple groups.