Hauck, Peter; Kurzweil, Hans A lattice theoretic characterization of prefrattini subgroups. (English) Zbl 0691.20016 Manuscr. Math. 66, No. 3, 295-301 (1990). 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 \({\mathcal G}\) [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. Reviewer: W.E.Deskins Cited in 3 Documents MSC: 20D30 Series and lattices of subgroups 20D25 Special subgroups (Frattini, Fitting, etc.) 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks 20D45 Automorphisms of abstract finite groups Keywords:subgroups; lattice; complemented; finite solvable group; minimal elements; prefrattini subgroups Citations:Zbl 0109.01403 PDFBibTeX XMLCite \textit{P. Hauck} and \textit{H. Kurzweil}, Manuscr. Math. 66, No. 3, 295--301 (1990; Zbl 0691.20016) Full Text: DOI EuDML References: [1] GASCHÜTZ, W.: Präfrattinigruppen. Archiv Math.13 (1962), 418–426 · Zbl 0109.01403 · doi:10.1007/BF01650090 [2] KURZWEIL, H.: A combinatorial technique for simplicial complexes and some applications to finite groups. To appear in Discrete Math. · Zbl 0727.20016 [3] KURZWEIL, H.: Die Praefrattinigruppe im Intervall eines Untergruppenverbands. Archiv Math.53 (1989), 235–244 · Zbl 0683.20019 · doi:10.1007/BF01277056 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.