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Chief factors, crowns, and the generalized Jordan-Hölder theorem. (English) Zbl 0649.20020

The paper deals with a new definition of the notion of “crown”, introduced by W. Gaschütz [Arch. Math. 13, 418-426 (1962; Zbl 0109.01403)] associated with a complemented chief factor H/K of a finite soluble group G and generalized for a non-Frattini chief factor of an arbitrary finite group by R. Baer, P. Förster and, independently, by J. Lafuente. This new definition has the advantage of being more esthetically pleasing, it significantly simplifies the proofs of the basic results related to crowns and also leads to a different approach to the generalized Jordan-Hölder Theorem.
Reviewer: R.Covaci

MSC:

20D30 Series and lattices of subgroups
20D40 Products of subgroups of abstract finite groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks

Citations:

Zbl 0109.01403
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References:

[1] Baer R., Illinois J. Math 1 pp 115– (1957)
[2] Baer R., Einbettungsrelationen und Formationen endlicher Gruppen
[3] DOI: 10.1016/0021-8693(68)90027-6 · Zbl 0177.03902 · doi:10.1016/0021-8693(68)90027-6
[4] DOI: 10.1007/BF01161802 · Zbl 0544.20015 · doi:10.1007/BF01161802
[5] DOI: 10.1007/BF01650090 · Zbl 0109.01403 · doi:10.1007/BF01650090
[6] DOI: 10.1017/S144678870001510X · Zbl 0298.20014 · doi:10.1017/S144678870001510X
[7] DOI: 10.1016/0021-8693(86)90058-X · Zbl 0591.20020 · doi:10.1016/0021-8693(86)90058-X
[8] DOI: 10.1017/S0305004100055262 · Zbl 0386.20008 · doi:10.1017/S0305004100055262
[9] DOI: 10.1007/BF01198125 · Zbl 0509.20011 · doi:10.1007/BF01198125
[10] DOI: 10.1080/00927878508823183 · Zbl 0556.20017 · doi:10.1080/00927878508823183
[11] DOI: 10.1080/00927878508823263 · Zbl 0575.20020 · doi:10.1080/00927878508823263
[12] Neumann H., Varieties of groups (1967) · Zbl 0149.26704
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