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Groups with many characteristically simple subgroups

To Philip Hall on his 75th Birthday

Published online by Cambridge University Press:  24 October 2008

J. S. Wilson
J. S. Wilson
Affiliation:
Christ's College, Cambridge
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Extract

1. A group G is called characteristically simple if it has no proper non-trivial subgroups which are invariant under all automorphisms of G. It is known that if G is characteristically simple then each countable subgroup lies in a countable characteristically simple subgroup of G. A similar assertion holds for simple groups. These results were proved by Philip Hall in lectures in 1966, and further proofs appear in (4) and (6). For simple groups there is a well known and elementary result in the other direction: if every two-generator subgroup of a group G lies in a simple subgroup, then G is simple. These considerations prompt the question (first raised, I believe, by Philip Hall) whether a group G is necessarily characteristically simple if each countable subgroup lies in a characteristically simple subgroup.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 193 - 198
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

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