On Deskins's conjecture (Q1380009)

Let \(G\) be a finite group and let \(M\) be a maximal subgroup of \(G\). A completion of \(M\) in \(G\) is a subgroup \(C\) of \(G\) such that \(C\nless M\) and every proper subgroup of \(C\) which is normal in \(G\) is contained in \(M\). The index complex of \(M\) in \(G\) is the set \(I(M)\) of all completions of \(M\) in \(G\) and \(P(M)=\{C\in I(M)\mid C\) is maximal in \(I(M)\) and \(G=CM\}\). \textit{W. E. Deskins} [Arch. Math. 54, No. 3, 236-240 (1990; Zbl 0665.20008)]\ made the conjecture that \(G\) is supersoluble iff for every maximal subgroup \(M\) of \(G\) the set \(P(M)\) contains an element \(C\) with \(C/K(C)\) cyclic, where \(K(C)\) is the product of all normal subgroups of \(G\) which are proper subgroups of \(C\), the so-called strict core of \(C\) in \(G\). The conjecture is not true: \(G=S_4\) is a counterexample, as was pointed out by \textit{A. Ballester-Bolinches} and \textit{L. M. Ezquerro} [Proc. Am. Math. Soc. 114, No. 2, 325-330 (1992; Zbl 0747.20008)]. In the paper under review, the author proves the following: Theorem. If \(G\) is \(S_4\)-free, then \(G\) is supersoluble if and only if for each maximal subgroup \(M\) of \(G\), \(P(M)\) contains an element \(C\) with \(C/K(C)\) cyclic.