A lattice theoretic characterization of prefrattini subgroups
If \({\mathcal U}\) is a subgroup of group \({\mathcal G}\), the set of all subgroups of \({\mathcal G}\) which contain \({\mathcal U}\), ordered by inclusion, is a lattice, \([{\mathcal U},{\mathcal G}]\). This lattice is complemented if for each \({\mathcal A}\) in \([{\mathcal U},{\mathcal G}]\) there exists \({\mathcal B}\) in [\({\mathcal U},{\mathcal G}]\) such that \({\mathcal A}\cap {\mathcal B}={\mathcal U}\) and \(<{\mathcal A},{\mathcal B}>={\mathcal G}\). Let \({\mathcal H}\) be a subgroup of finite solvable group \({\mathcal G}\) and denote by \(L=L({\mathcal H},{\mathcal G})\) the set \(\{{\mathcal U}\) in \([{\mathcal H},{\mathcal G}]:\) \([{\mathcal U},{\mathcal G}]\) is complemented\}. The authors prove by induction that the minimal elements of L are conjugate in \({\mathcal G}\) (but an example is given to show that they are not necessarily conjugate in the normalizer of \({\mathcal H})\). From work of the second author it follows that these are the \({\mathcal H}\)-prefrattini subgroups and, for \({\mathcal H}=1\), the prefrattini subgroups of \({\mathcal G}\) [\textit{W. Gaschütz}, Arch. Math. 13, 418-426 (1962; Zbl 0109.01403)]. In Section 4 an example is given of a nonsolvable group \({\mathcal G}\) having a subgroup \({\mathcal H}\) for which some minimal elements of L(\({\mathcal H},{\mathcal G})\) have different orders.
scientific article; zbMATH DE number 4130650