First-order characterization of the radical of a finite group
John S. Wilson
Source: J. Symbolic Logic Volume 74, Issue 4 (2009), 1429-1435.
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.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748698
Digital Object Identifier: doi:10.2178/jsl/1254748698
References
M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Mathematical Journal, vol. 63 (1976), pp. 1--91.
Mathematical Reviews (MathSciNet):
MR422401
Project Euclid: euclid.nmj/1118795948
R. W. Carter, Simple groups of Lie type, John Wiley & Sons, London--New York--Sydney, 1989.
Mathematical Reviews (MathSciNet):
MR1013112
J. Dieudonné, La géometrie des groupes classiques, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 5, Springer-Verlag, Berlin--Göttingen--Heidelberg, 1970.
N. Gordeev, F. Grunewald, B. Kunyavski\u i, and E. Plotkin, A commutator description of the solvable radical of a finite group, Groups, geometry and dynamics, vol. 2, 2008, pp. 85--120.
Mathematical Reviews (MathSciNet):
MR2367209
Zentralblatt MATH:
1161.20016
D. Gorenstein, Finite simple groups. An introduction to their classification, Plenum Publishing Corporation, New York, 1982.
Mathematical Reviews (MathSciNet):
MR698782
R. M. Guralnick, B. Kunyavski\u i, E. Plotkin, and A. Shalev, Thompson-like characterizations of the solvable radical, Journal of Algebra, vol. 300 (2006), pp. 363--375.
Mathematical Reviews (MathSciNet):
MR2228653
Zentralblatt MATH:
1118.20021
Digital Object Identifier: doi:10.1016/j.jalgebra.2006.03.001
N. Nikolov and D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Annals of Mathemathics. Second Series, vol. 165 (2007), pp. 171--238.
Mathematical Reviews (MathSciNet):
MR2276769
Zentralblatt MATH:
1126.20018
Digital Object Identifier: doi:10.4007/annals.2007.165.171
F. Point, Ultraproducts and Chevalley groups, Archive for Mathematical Logic, vol. 38 (1999), pp. 355--372.
Mathematical Reviews (MathSciNet):
MR1711404
Zentralblatt MATH:
0921.03008
Digital Object Identifier: doi:10.1007/s001530050131
J. S. Wilson, Finite axiomatization of finite soluble groups, Journal of the London Mathematical Society, vol. 74 (2006), no. 2, pp. 566--582.
Mathematical Reviews (MathSciNet):
MR2286433
Zentralblatt MATH:
1118.20019
Digital Object Identifier: doi:10.1112/S0024610706023106
--------, Characterization of the soluble radical by a sequence of words, Journal of Algebra,(to appear).
