On Deskins's conjecture concerning the supersolvability of a finite group (Q1380024)

Let \(G\) be a finite group, let \(M\) be a maximal subgroup of \(G\) and let \(I(M)\) denote the index complex of \(M\) in \(G\) [see the preceding review Zbl 0893.20007]. The author calls a subgroup \(C\) of \(G\) a \(\theta\)-subgroup for \(M\) if \(C\nless M\) and \(\text{core}_G(M\cap C)\) is maximal among the proper normal subgroups of \(G\) contained in \(C\). The set of all \(\theta\)-subgroups for \(M\) is denoted by \(\theta(M)\) and the maximal elements of \(\theta(M)\) with respect to inclusion are called maximal \(\theta\)-subgroups for \(M\). The main result of the paper is the following: Theorem. Let \(G\) be a finite group and suppose that for every maximal subgroup \(M\) of composite index of \(G\) there exists a maximal \(\theta\)-subgroup \(C\) for \(M\) such that \(G=CM\) and \(G/\text{core}_G(M\cap C)\) is cyclic. Then either \(G\) is supersoluble or else \(G\) has a homomorphic image isomorphic to \(S_4\). Remark. If \(C\in I(M)\), then \(C\) is a \(\theta\)-subgroup for \(M\) and \(\text{core}_G(M\cap C)=K(C)\); it follows that \(I(M)\subseteq\theta(M)\). Therefore Zhao's result is a slight extension of Guo's theorem described in the preceding review.