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On characteristically simple groups

Published online by Cambridge University Press:  24 October 2008

J. S. Wilson
J. S. Wilson
Affiliation:
Christ's College, Cambridge
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1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 1 , July 1976 , pp. 19 - 35
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Artin, E.Geometric algebra (New York and London, Interscience, 1957).Google Scholar
(2)Bass, H.Algebraic K-theory (New York and Amsterdam, Benjamin, 1968).Google Scholar
(3)Bass, H., NOR, J. and Serre, J.-P.Solution of the congruence subgroup problem for SLn (n ≥ 3) and Sp2n (n ≥ 2). Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59–137.CrossRefGoogle Scholar
(4)Brenner, J. L.The linear homogeneous group. III. Ann. of Math. (2) 71 (1960), 210–223.CrossRefGoogle Scholar
(5)Camm, R.Simple free products. J. London Math. Soc. 28 (1953), 66–76.CrossRefGoogle Scholar
(6)Cohn, P. M.Free rings and their relations (London and New York, Academic Press, 1971).Google Scholar
(7)Hall, P.Wreath powers and characteristically simple groups. Proc. Cambridge Philos. Soc. 58 (1962), 170–184.CrossRefGoogle Scholar
(8)Hall, P.On the embedding of a group in a join of given groups. J. Austral. Math. Soc. 17 (1974), 434–495.Google Scholar
(9)Kourov notebook. Unsolved problems in the theory of groups. Fourth augmented edition. Academy of Sciences of the U.S.S.R., Siberian Branch, Institute of Mathematics, 1973.Google Scholar
(10)Kovács, L. G. and NEwMAx, M. F.Minimal verbal subgroups. Proc. Cambridge Philos. Soc. 62 (1966), 347–350.CrossRefGoogle Scholar
(11)Litoff, O.On the commutator subgroup of the general linear group. Proc. Amer. Math. Soc. 6 (1955), 465–470.Google Scholar
(12)McLain, D. H.A characteristically-simple group. Proc. Cambridge Philos. Soc. 50 (1954), 641–642.CrossRefGoogle Scholar
(13)McLain, D. H. A class of locally nilpotent groups. Cambridge dissertation, 1956.CrossRefGoogle Scholar
(14)Neumann, B. H. and Yamamuro, S.Boolean powers of simple groups. J. Austral. Math. Soc. 5 (1965), 315–324.CrossRefGoogle Scholar
(15)Robertson, E. F.A remark on the derived group of GL(R). Bull. London Math. Soc. 1 (1969), 160–162.CrossRefGoogle Scholar
(16)Scureier, O. and Van Der Waerden, B. L.Die Automorphismen der projektiven Gruppen. Abh. Math. Sem. Univ. Hamburg 6 (1928), 303–322.CrossRefGoogle Scholar
(17)Tyrer, Jos J. M.Direct products and the Hopf property. J. Austral. Math. Soc. 17 (1974), 174–196.Google Scholar
(18)Wilson, J. S. Some properties of groups inherited by normal subgroups of finite index.Math. Z. 114 (1970), 19–21.CrossRefGoogle Scholar
(19)Wilson, J. S.Groups satisfying the maximal condition for normal subgroups. Math. Z. 118 (1970), 107–114.Google Scholar
(20)Wilson, J. S.The normal and subnormal structure of general linear groups. Proc. Cambridge Philos. Soc. 71 (1972), 163–177.CrossRefGoogle Scholar