Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-19T16:58:00.916Z Has data issue: false hasContentIssue false
  • English
  • Français

The Jordan-Hölder theorem and prefrattini subgroups of finite groups

Published online by Cambridge University Press:  18 May 2009

A. Ballester-Bolinches  and
L. M. Ezquerro
A. Ballester-Bolinches
Affiliation:
Departament D'Àlgebra, Universitat de València, C/Dr. Moliner 50, 46100 Burjassot (València), Spain.
L. M. Ezquerro
Affiliation:
Departamento de Mátematica E Informática, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 3 , September 1995 , pp. 265 - 277
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

REFERENCES

1.Ballester-Bolinches, A. and Ezquerro, L. M., On maximal subgroups of finite groups, Comm. Algebra 19(8) (1991), 2373–2394.CrossRefGoogle Scholar
2.Barnes, D. W., On complemented chief factors of finite soluble groups, Bull. Austral. Math. Soc. 7 (1972), 101–104.Google Scholar
3.Bechtell, H., Pseudo-Frattini subgroups, Pacific J. Math. 14 (1964), 1129–1136.CrossRefGoogle Scholar
4.Brandis, A., Moduln und verschrankte Homomorphismen endlicher Gruppen, J. reine angew. Math. 385 (1988), 102–116.Google Scholar
5.Carter, R. W., Fischer, B. and Hawkes, T. O., Extreme classes of finite soluble groups, J. Algebra 9 (1968), 285–313.CrossRefGoogle Scholar
6.Deskins, W. E., On maximal subgroups, Proc. Symp. in Pure Math., Amer. Math. Soc. 1 (1959), 100–104.Google Scholar
7.Doerk, K. and Hawkes, T. O., Finite Soluble Groups, (de Gruyter, 1992).CrossRefGoogle Scholar
8.Förster, P., Prefrattini subgroups, J. Austral. Math. Soc. (Series A) 34 (1983), 234–247.CrossRefGoogle Scholar
9.Förster, P., Chief factors, crowns and the generalised Jordan-Hölder Theorem, Comm. Algebra 16(8) (1988), 1627–1638.CrossRefGoogle Scholar
10.Gaschütz, W., Praefrattinigruppen, Arch. Math. 13 (1962), 418–426.CrossRefGoogle Scholar
11.Hawkes, T. O., Analogues of Prefrattini subgroups, Proc. Internal. Conf. Theory of Groups, Austral. Nat. Univ. Canberra. (1965), 145–150.Google Scholar
12.Kurzweil, H., Die Praefrattinigruppe in Intervall eines Untergruppenverbandes, Arch. Math. 53 (1989), 235–244.CrossRefGoogle Scholar
13.Lafuente, J., Maximal subgroups and the Jordan-Holder Theorem, J. Austral. Math. Soc. (Series A) 46 (1989), 356–364.CrossRefGoogle Scholar
14.Tomkinson, M. J., Prefrattini subgroups and cover-avoidance properties in II-groups, Can. J. Math. 27 (1975), 837–851.Google Scholar