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Fitting functors in finite solvable groups. II. (English) Zbl 0632.20013

This paper is a continuation of Part I [Math. Z. 182, 359-384 (1983; Zbl 0518.20014)]. It mainly intends to study the behaviour of Fitting functors with respect to direct products. Lockett functors are introduced: they are Fitting functors f having the property that in any direct product \(G\times G\), G being soluble, each member in f(G\(\times G)\) splits into a direct product of two groups in f(G). It is shown that every Fitting functor satisfying the Frattini argument is strongly contained in a Lockett functor, and amongst these Lockett functors there is a unique smallest, the “upper star” of the Fitting functor. In an attempt to generalize a result about Lockett sections of Fitting classes of F. P. Lockett [Math. Z. 137, 131-136 (1974; Zbl 0286.20017)], the authors consider the question whether Fitting functors have “lower stars” and obtain a partial answer.
Reviewer: H.Lausch

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

[1] DOI: 10.1007/BF01214854 · Zbl 0286.20017 · doi:10.1007/BF01214854
[2] DOI: 10.1112/jlms/s2-20.3.423 · Zbl 0422.20017 · doi:10.1112/jlms/s2-20.3.423
[3] DOI: 10.1007/BF01179756 · Zbl 0518.20014 · doi:10.1007/BF01179756
[4] DOI: 10.1007/BF01235351 · Zbl 0447.20019 · doi:10.1007/BF01235351
[5] DOI: 10.1007/BF01222717 · Zbl 0405.20022 · doi:10.1007/BF01222717
[6] DOI: 10.1112/plms/s3-19.2.193 · Zbl 0169.34201 · doi:10.1112/plms/s3-19.2.193
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