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Groups sharing some varietal properties with supersoluble groups

  • Rolf Brandl (a1)

Abstract

In this note a formation U is considered which can be defined by a sequence of laws which ‘almost’ hold in every finite supersoluble group. The class U contains all finite supersoluble groups and each group in U has a Sylow tower.

It is shown that a finite group belongs to U if and only if all of its subgroups with nilpotent commutator subgroup are supersoluble. A more general result concerning classes of this type finally proves that U is a saturated formation.

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References

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[1]Brandi, R., ‘Zur Theone der untergruppenabgeschlossenen Formationen: Endliche Varietäten,’ J. Algebra 73 (1981), 122.
[2]Carter, R., Fischer, B. and Hawkes, T., ‘Extreme classes of finite groups,’ J. Algebra 9 (1968), 285313.
[3]Doerk, K., ‘Zur Theorie der Formationen endlicher auflösbarer Gruppen,’ J. Algebra 13 (1969). 345373.
[4]Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, (1967).
[5]Napolitani, F., ‘Sui gruppi finiti privi di sottogruppi supersolubili non speciali,’ Rend. Sem. Mat. Univ. Padova 39 (1967), 291295.
[6]Scarselli, A., ‘Sui gruppi a sottogruppi supersolubili abeliani,’ Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 60 (1976), 564569.
[7]Scarselli, A., ‘SA-gruppi e SN-gruppi,’ Boll. Un. Mat. Ital. A (5) 14 (1977), 174182.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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