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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
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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

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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