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A lattice theoretic characterization of prefrattini subgroups. (English) Zbl 0691.20016

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

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

Citations:

Zbl 0109.01403
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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
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