Some solubility criteria in factorised groups. (Q2907018)

Let the finite group \(G=AB\) be the product of two subgroups \(A\) and \(B\). If \(A\) and \(B\) are Abelian then, by a well known theorem of \textit{N. Itô} [Math. Z. 62, 400-401 (1955; Zbl 0064.25203)], \(G\) is metabelian. Unfortunately the results obtained later do not have the same generality and elegance as Itô's theorem. Many authors have therefore been forced to add some hypotheses (such as, for example, that \(A\) and \(B\) are totally permutable, namely for each subgroup \(A_0\) of \(A\) and for each subgroup \(B_0\) of \(B\) to have \(A_0B_0=B_0A_0\)) in order to obtain some information about the structure of \(G\).NEWLINENEWLINE The main result of the paper under review is: Assume that a finite group \(G=AB\) is the product of the soluble subgroups \(A\) and \(B\) and let \(p=\min\pi(G)\). If \((|A|,|B|)=1\), \(p\in\pi(A)\), and \(B\) permutes with every maximal subgroup of \(A\), then \(G\) is soluble.