Variations on results on orders of products in finite groups

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.

On the supersolvable residual of an M-group

Using the technique of linear limits of characters due to Dade and Loukaki, we give some conditions on the supersolvable residual of a finite solvable group [Formula: see text] that is sufficient to guarantee that [Formula: see text] is an [Formula: see text]-group. The monomiality of normal subgroups and Hall subgroups of the group [Formula: see text] are also determined.