A solubility criterion for finite groups. (Q2487033)

Let \(G\) be a finite group. A subgroup \(B\) of \(G\) is called a supplement of the subgroup \(A\) of \(G\) if \(G=AB\). The authors prove that \(G\) is soluble if the normalizer of every cyclic subgroup of prime-power order of \(G\) has a soluble supplement. This theorem is a generalization of a recent result of \textit{Y. Berkovich} and \textit{L. Kazarin} [J. Algebra 283, No. 2, 564-583 (2005; Zbl 1112.20021)].