Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T16:05:38.974Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgroups like Wielandt's in finite soluble groups

Published online by Cambridge University Press:  24 October 2008

R. A. Bryce
R. A. Bryce
Affiliation:
Department of Mathematics, Faculty of Science, Australian National University, P.O. Box 4, Canberra A.C.T. 2601, Australia
Get access Rights & Permissions [Opens in a new window]

Extract

In 1935 Baer[1] introduced the concept of kern of a group as the subgroup of elements normalizing every subgroup of the group. It is of interest from three points of view: that of its structure, the nature of its embedding in the group, and the influence of its internal structure on that of the whole group. The kern is a Dedekind group because all its subgroups are normal. Its structure is therefore known exactly (Dedekind [7]): if not abelian it is a direct product of a copy of the quaternion group of order 8 and an abelian periodic group with no elements of order 4. As for the embedding of the kern, Schenkman[13] shows that it is always in the second centre of the group: see also Cooper [5], theorem 6·5·1. As an example of the influence of the structure of the kern on its parent group we cite Baer's result from [2], p. 246: among 2-groups, only Hamiltonian groups (i.e. non-abelian Dedekind groups) have nonabelian kern.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 2 , March 1990 , pp. 239 - 259
Copyright
Copyright © Cambridge Philosophical Society 1990

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]Baer, R.. Der Kern, eine charakteristische Untergruppe. Compositio Math. 1 (1935), 254–283.Google Scholar
[2]Baer, R.. Gruppen mit Hamiltonschen Kern. Compositio Math. 2 (1935), 241–246.Google Scholar
[3]Bryce, R. A. and Cossey, John. The Wielandt subgroup of a finite soluble group. J. London Math. Soc. (to appear).Google Scholar
[4]Camina, A. R.. The Wielandt length of finite groups. J. Algebra 15 (1970), 142–148.CrossRefGoogle Scholar
[5]Cooper, C.. Power automorphisms of a group. Math. Z. 107 (1968), 335–356.Google Scholar
[6]Curtis, C. W. and Reiner, I.. Representation Theory of Finite Groups and Associative Algebras (Interscience Publishers, 1962).Google Scholar
[7]Dedekind, R.. Üher Gruppen deren sämtliche Teiler Normalteiler sind. Math. Ann. 48 (1897), 548–561.CrossRefGoogle Scholar
[8]Gaschütz, W.. Gruppen in denen das Normalteilersein transitiv ist. J. Reine Angew. Math. 198 (1957), 87–92.CrossRefGoogle Scholar
[9]Huppert, B.. Endliche Gruppen I. Grundlehren Math. Wiss. Band 134 (Springer-Verlag, 1967).Google Scholar
[10]Neumann, H.. Varieties of Groups. Ergeb. Math. Grenzgeb. Band 37 (Springer-Verlag, 1967).CrossRefGoogle Scholar
[11]Robinson, D. J. S.A note on finite groups in which normality is transitive. Proc. Anier. Math. Soc. 19 (1968), 933–937.CrossRefGoogle Scholar
[12]Robinson, D. J. S.. A Course in the Theory of Groups. Graduate Texts in Math, no. 80 (Springer-Verlag, 1982).CrossRefGoogle Scholar
[13]Schenkman, E.. On the norm of a group. Illinois J. Math. 4 (1960), 150–152.CrossRefGoogle Scholar
[14]Wielandt, H.. Uber den Normalisator der subnormalen Untergruppen. Math. Z. 69 (1958), 463–465.CrossRefGoogle Scholar
[15]Zacher, G.. Caratterizzazione dei t-gruppi finiti risolubili. Ricerche Mat. 1 (1952), 287–294.Google Scholar