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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

(1)Cantor, G.Beiträge zur Begründung der transfiniten Mengenlehre: I. Math. Ann. 46 (1895), 481–512.CrossRefGoogle Scholar
(2)Hall, P.Wreath powers and characteristically simple groups. Proc. Cambridge Philos. Soc. 58 (1962), 170–184.CrossRefGoogle Scholar
(3)Hausdorff, F.Set Theory, 3rd ed. (New York, Chelsea, 1962).Google Scholar
(4)Kopperman, R. D. and Mathias, A. R. D. Some problems in group theory. In The syntax and semantics of infinitary languages, 131–138. Lecture Notes in Mathematics, no. 72 (Berlin, Heidelberg, Springer-Verlag, 1968).CrossRefGoogle Scholar
(5)McLain, D. H.A characteristically simple group. Proc. Cambridge Philos. Soc. 50 (1954), 641–642.CrossRefGoogle Scholar
(6)Phillips, R. E.Countably recognizable classes of groups. Rocky Mtn J. of Math. 1 (1971), 489–497.CrossRefGoogle Scholar
(7)Robinson, D. J. S.Finiteness conditions and generalized soluble groups, Part II. Ergebnisse der Mathematik und ihre Grenzgebiete, no. 63 (Berlin, Heidelberg, New York, Springer-Verlag, 1972).CrossRefGoogle Scholar
(8)Roseblade, J. E.The automorphism group of McLain's characteristically simple group. Math. Z. 82 (1963), 267–282.CrossRefGoogle Scholar