Sufficient conditions for the solvability of a finite group (Q2418201)

A famous theorem of [\textit{H. Wielandt}, Ill. J. Math. 2, 611--618 (1958; Zbl 0084.02904)] shows that every finite group which is factorized by two nilpotent subgroups of coprime orders is soluble. Wielandt's result [loc. cit.] was later improved by \textit{O. H. Kegel} [Arch. Math. 12, 90--93 (1961; Zbl 0099.01401)],who removed the coprime order assumption on the factors. In the paper under review, the authors provide the following interesting generalization of the theorem of Wielandt: if a finite group \(G=HK\) is the product of two subgroups \(H\) and \(K\) of coprime orders, then \(G\) is soluble, provided that \(H\) is \(2\)-nilpotent and \(K\) is nilpotent of odd order. The authors leave as an open question whether the coprimality assumption can be omitted in this case. A suitable easy example shows the \(2\)-nilpotency assumption on \(H\) cannot be replaced by \(p\)-nilpotency for a prime number \(p\) which does not divide the order of \(K\).