On maximal subgroups of finite groups. (English) Zbl 0798.20011
From the authors’ introduction: “Every group considered will be finite. The role of Hall systems in the theory of soluble groups is well known but nothing similar is known in the universe of all finite groups. In this paper we introduce something we call systems of maximal subgroups which can be used to ‘select’ maximal subgroups in order to introduce prefrattini subgroups similar to those of W. Gaschütz [Arch. Math. 13, 418–426 (1962; Zbl 0109.01403)]. These prefrattini subgroups enjoy most of the properties of the soluble universe except the cover and avoidance properties and although they are not conjugate (in fact conjugacy characterizes solubility) their core is always the Frattini subgroup.
Our approach to prefrattini subgroups includes the classical Hawkes’ and Förster’s introductions [in T. Hawkes, Proc. Int. Conf., Aust. Natl. Univ., Canberra, 145–150 (1965; Zbl 0183.03002) and P. Förster, J. Aust. Math. Soc., Ser. A 34, 234–247 (1983; Zbl 0519.20016)] even the celebrated Hawkes factorization of the \(F\)-prefrattini subgroups, when \(F\) is a saturated formation, despite the (fact that) cover and avoidance properties do not hold here.
Finally we use our prefrattini subgroups to characterize the so-called totally unsaturated formations [cf. M. C. Hofmann, Arch. Math. 48, 199–207 (1987; Zbl 0607.20008)]”.
Our approach to prefrattini subgroups includes the classical Hawkes’ and Förster’s introductions [in T. Hawkes, Proc. Int. Conf., Aust. Natl. Univ., Canberra, 145–150 (1965; Zbl 0183.03002) and P. Förster, J. Aust. Math. Soc., Ser. A 34, 234–247 (1983; Zbl 0519.20016)] even the celebrated Hawkes factorization of the \(F\)-prefrattini subgroups, when \(F\) is a saturated formation, despite the (fact that) cover and avoidance properties do not hold here.
Finally we use our prefrattini subgroups to characterize the so-called totally unsaturated formations [cf. M. C. Hofmann, Arch. Math. 48, 199–207 (1987; Zbl 0607.20008)]”.
Reviewer: W. E. Deskins (Pittsburgh)
MSC:
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20E28 | Maximal subgroups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
Keywords:
Hall systems; finite groups; systems of maximal subgroups; prefrattini subgroups; cover and avoidance; core; Frattini subgroup; Hawkes factorization; formationsReferences:
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