Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T08:34:20.159Z Has data issue: false hasContentIssue false
  • English
  • Français

On groups satisfying the converse of Lagrange's theorem

Published online by Cambridge University Press:  24 October 2008

J. F. Humphreys
J. F. Humphreys
Affiliation:
Department of Pure Mathematics, University of Liverpool
Get access Rights & Permissions [Opens in a new window]

Extract

In this article we study certain subclasses of the class ℒ of Lagrangian groups; that is, finite groups G having, for every divisor d of |G|, a subgroup of index d. Two such subclasses, mentioned by McLain in (6), are the class ℒ1 of groups G such that every factor group of G is in ℒ, and the class ℒ2 of groups G such that each subnormal subgroup of G is in ℒ. In section 1 we prove that a group of odd order in ℒ1 is supersoluble, and give some examples of non-supersoluble groups in ℒ1. Section 2 contains several results on the class ℒ2. In particular, it is shown that a group in ℒ2 has an ordered Sylow tower and, after constructing some examples of groups in ℒ2, a result on the rank of a group in ℒ2 is proved (Theorem 4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 1 , January 1974 , pp. 25 - 32
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Dixon, J. D.Normal p–subgroups of solvable linear groups. J. Austral. Math. Soc. 7 (1967), 545–551.CrossRefGoogle Scholar
(2)Humphreys, J. F. and Johnson, D. L.On Lagrangian groups. Trans. Amer. Math. Soc. 181 (1973).Google Scholar
(3)Huppert, B.Lineare auflösbare Gruppen. Math. Z. 67 (1957), 479–518.CrossRefGoogle Scholar
(4)Huppert, B.Endliche Gruppen I (Berlin, Springer-Verlag, 1967).CrossRefGoogle Scholar
(5)Joson, D. L.A note on supersoluble groups. Canad. J. Math. 23 (1971), 562–564.Google Scholar
(6)McLain, D. H.The existence of subgroups of given order in finite groups. Proc. Cambridge Philos. Soc. 53 (1957), 278–285.CrossRefGoogle Scholar