On \(\theta\)-pairs for maximal subgroups (Q1973361)

A pair of subgroups \((C,D)\) of a finite group \(G\) is said to be a \(\theta^*\)-pair for a maximal subgroup \(M\) of \(G\) if it satisfies the following properties: (a) \(D\) is a proper subgroup of \(C\) and \(D\) is normal in \(G\). (b) \(D\) is contained in \(M\) and \(M\) does not contain any conjugate of \(C\) in \(G\). (c) \(C/D\) has no proper normal subgroup of \(G/D\). A \(\theta^*\)-pair \((C,D)\) for \(M\) is said to be maximal if \(M\) has no \(\theta^*\)-pair \((C',D')\) such that \(C<C'\). In this paper the authors impose conditions on maximal \(\theta^*\)-pairs which imply either solubility or supersolubility. Earlier results on \(\theta\)-pairs are improved. Some theorems included in the paper are the following: Theorem 1. A finite group \(G\) is soluble if and only if for each maximal subgroup of composite index, there exists a maximal \(\theta^*\)-pair \((C,D)\) such that \(C/D\) is nilpotent. Theorem 2. Let \(G\) be a finite group with a nilpotent maximal subgroup \(M\) such that \(G=MF(G)\). Assume that \(M\) has a maximal \(\theta^*\)-pair \((C,D)\) such that \(C/D\) is cyclic. Then \(G\) is supersoluble. The last theorems contain some known results on saturated formations.