On \(\widehat\theta\)-pairs for maximal subgroups of a finite group. (Q2856528)

This paper deals with the influence of the maximal subgroups on the structure of finite groups. The following concept is introduced: For a maximal subgroup of a group \(G\), a \(\widehat\theta\)-pair for \(M\) is any pair of subgroups \((C,D)\) of \(G\) such that (i) \(D\) is normal in \(G\), (ii) \(\langle M,D\rangle=M\), (iii) no non-trivial proper subgroup of \(C/D\) is normal in \(G/D\), and (iv) either \(C=G\) or there exists a subgroup \(E\) of \(G\) such that \(C\) is a maximal subgroup of \(E\) and \((EM_G)_G\) is not contained in \(M\). (Here \(X_G\) denotes the core of the subgroup \(X\) in \(G\).)NEWLINENEWLINE This notion is related with the ones of completion [\textit{W. E. Deskins}, Arch. Math. 54, No. 3, 236-240 (1990; Zbl 0665.20008)] and \(\theta\)-pair [\textit{N. P. Mukherjee} and \textit{P. Bhattacharya}, Proc. Am. Math. Soc. 109, No. 3, 589-596 (1990; Zbl 0699.20019)].NEWLINENEWLINE By using \(\widehat\theta\)-pairs, new characterizations for a finite group to be soluble, supersoluble, nilpotent, etc. are presented in the paper.