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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

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