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First-order characterization of the radical of a finite group

Published online by Cambridge University Press:  12 March 2014

John S. Wilson
John S. Wilson*
Affiliation:
University College, Oxford Ox1 4Bh, UK, E-mail: [email protected]
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Abstract

It is shown that there is a formula σ(g) in the first-order language of group theory with the following property: for every finite group G, the largest soluble normal subgroup of G consists precisely of the elements g of G such that σ(g) holds.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1429 - 1435
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1]Aschbacher, M. and Seitz, G. M., Involutions in Chevalley groups over fields of even order, Nagoya Mathematical Journal, vol. 63 (1976), pp. 1–91.CrossRefGoogle Scholar
[2]Carter, R. W., Simple groups of Lie type, John Wiley & Sons, London–New York–Sydney, 1989.Google Scholar
[3]Dieudonné, J., La géometrie des groupes classiques, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 5, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1970.Google Scholar
[4]Gordeev, N., Grunewald, F., Kunyavskiǐ, B., and Plotkin, E., A commutator description of the solvable radical of a finite group, Groups, geometry and dynamics, vol. 2, 2008, pp. 85–120.CrossRefGoogle Scholar
[5]Gorenstein, D., Finite simple groups. An introduction to their classification, Plenum Publishing Corporation, New York, 1982.Google Scholar
[6]Guralnick, R. M., Kunyavskiǐ, B., Plotkin, E., and Shalev, A., Thompson-like characterizations of the solvable radical, Journal of Algebra, vol. 300 (2006), pp. 363–375.CrossRefGoogle Scholar
[7]Nikolov, N. and Segal, D., On finitely generated profinite groups. I. Strong completeness and uniform bounds, Annals of Mathemathics. Second Series, vol. 165 (2007), pp. 171–238.CrossRefGoogle Scholar
[8]Point, F., Ultraproducts and Chevalley groups, Archive for Mathematical Logic, vol. 38 (1999), pp. 355–372.CrossRefGoogle Scholar
[9]Wilson, J. S., Finite axiomatization of finite soluble groups, Journal of the London Mathematical Society, vol. 74 (2006), no. 2, pp. 566–582.CrossRefGoogle Scholar
[10]Wilson, J. S., Characterization of the soluble radical by a sequence of words, Journal of Algebra, (to appear).Google Scholar