Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-27T16:17:10.612Z Has data issue: false hasContentIssue false

AN IDENTIFICATION THEOREM FOR GROUPS WITH SOCLE PSU${}_{6} (2)$

Published online by Cambridge University Press:  25 February 2013

CHRIS PARKER*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
GERNOT STROTH
Affiliation:
Institut für Mathematik, Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, 06099 Halle, Germany email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We identify the groups ${\text{PSU} }_{6} (2)$, ${\text{PSU} }_{6} (2){: }2$, ${\text{PSU} }_{6} (2){: }3$ and $\text{Aut} ({\text{PSU} }_{6} (2))$ from the structure of the centralizer of an element of order three.

Type
Research Article
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Aschbacher, M., Finite Group Theory, Cambridge Studies in Advanced Mathematics, 10 (Cambridge University Press, Cambridge, 1986).Google Scholar
Brauer, R. and Suzuki, M., ‘On finite groups of even order whose 2-Sylow group is a quaternion group’, Proc. Natl Acad. Sci. USA 45 (1959), 17571759.CrossRefGoogle Scholar
Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups (Clarendon Press, Oxford, 1985).Google Scholar
Feit, W. and Thompson, J. G., ‘Finite groups which contain a self-centralizing subgroup of order three’, Nagoya Math. J. 21 (1962), 185197.CrossRefGoogle Scholar
Goldschmidt, D., ‘2-fusion in finite groups’, Ann. of Math. 99 (1974), 70117.CrossRefGoogle Scholar
Gorenstein, D., Finite Groups (Harper & Row, New York, 1968).Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups, No. 2, Mathematical Surveys and Monographs, 40.2 (American Mathematical Society, Providence, RI, 1996).Google Scholar
Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups, No. 3, Mathematical Surveys and Monographs, 40.3 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Hayden, J. L., ‘A characterization of the finite simple group ${\mathrm{PSp} }_{4} (3)$’, Canad. J. Math. 25 (1973), 539553.CrossRefGoogle Scholar
Huppert, B., Endliche Gruppen (Springer, Berlin, 1967).CrossRefGoogle Scholar
Isaacs, I. M., Character Theory of Finite Groups (AMS Chelsea Publishing, Providence, RI, 2006).Google Scholar
Meierfrankenfeld, U., Stellmacher, B. and Stroth, G., ‘The structure theorem for finite groups with a large $p$-subgroup’, Preprint, 2011.Google Scholar
Parker, C., ‘A 3-local characterization of ${\mathrm{U} }_{6} (2)$ and ${\mathrm{Fi} }_{22} $’, J. Algebra 300 (2) (2006), 707728.CrossRefGoogle Scholar
Parker, C. and Rowley, P., ‘A 3-local identification of the alternating group of degree eight, the McLaughlin simple group and their automorphism groups’, J. Algebra 319 (2008), 17521775.CrossRefGoogle Scholar
Parker, C. and Rowley, P., ‘A 3-local characterization of ${\mathrm{Co} }_{2} $’, J. Algebra 323 (3) (2010), 601621.CrossRefGoogle Scholar
Parker, C. and Stroth, G., ‘Strongly $p$-embedded subgroups’, Pure Appl. Math. Q. 7(3 special issue: in honor of Jacques Tits) (2011), 797–858.CrossRefGoogle Scholar
Parker, C. and Stroth, G., ‘Groups which are almost groups of Lie type in characteristic $p$’, Preprint, arXiv:1110.1308.Google Scholar
Parker, C. and Stroth, G., ‘${\mathrm{F} }_{4} (2)$ and its automorphism group’, Preprint, arXiv:1108.1661.Google Scholar
Parker, C., Salarian, M. R. and Stroth, G., ‘A characterisation of ${\text{} }^{2} {\mathrm{E} }_{6} (2)$, $\mathrm{M} (22)$ and $\mathrm{Aut} (\mathrm{M} (22))$ from a characteristic 3 perspective’, Preprint, arXiv:1108.1894.Google Scholar
Parker, C. and Stroth, G., ‘An identification theorem for the sporadic simple groups ${\mathrm{F} }_{2} $ and $\mathrm{M} (23)$’, Preprint, arXiv:1201.3229v1.Google Scholar
Prince, A. R., ‘A characterization of the simple groups $\mathrm{PSp} (4, 3)$ and $\mathrm{PSp} (6, 2)$’, J. Algebra 45 (2) (1977), 306320.CrossRefGoogle Scholar
Seidel, A., Gruppen lokaler Charakteristik — eine Kennzeichnung von Gruppen vom Lie Typ in ungerader Charakteristik, Dissertation, Universität Halle-Wittenberg, 2009. http://digital.bibliothek.uni-halle.de/hs/content/titleinfo/397705.Google Scholar
Smith, F., ‘A characterization of the .2 Conway simple group’, J. Algebra 31 (1974), 91116.CrossRefGoogle Scholar
Yoshida, T., ‘Character-theoretic transfer’, J. Algebra 52 (1978), 138.CrossRefGoogle Scholar