Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T05:34:04.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite simple groups with no elements of order six

Published online by Cambridge University Press:  17 April 2009

L. M. Gordon
L. M. Gordon
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
Rights & Permissions [Opens in a new window]

Extract

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.

The aim of this paper is to completely classify the finite simple groups with no elements of order 6. The proof is by induction and involves an analysis of the structure of the 2-local subgroups of a counterexample of minimum order. A recent result of Glauberman plays an essential role.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 17 , Issue 2 , October 1977 , pp. 235 - 246
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Alperin, J.L., Brauer, Richard and Gorenstein, Daniel, “Finite groups with quasidihedral and wreathed Sylow 2-subgroups”, Trans. Amer. Math. Soc. 151 (1970), 1–26.Google Scholar
[2]Alperin, J.L. and Gorenstein, Daniel, “The multiplicators of certain simple groups”, Proc. Amer. Math. Soc. 17 (1966), 515–519.CrossRefGoogle Scholar
[3]Dempwolff, Ulrich, “A characterization of the Rudvalis simple group of order 214.33.53.7.13.29 by the centralizers of noncentral involutions”, J. Algebra 32 (1974), 53–88.CrossRefGoogle Scholar
[4]Fletcher, L.R., Stellmacher, B., Stewart, W.B., “Endliche Gruppen, die kein Element der Ordung 6 enthalten”, (unpublished).Google Scholar
[5]Gilman, Robert and Gorenstein, Daniel, “Finite groups with Sylow 2-subgroups of class two. I”, Trans. Amer. Math. Soc. 207 (1975), 1–101.CrossRefGoogle Scholar
[6]Gilman, Robert and Gorenstein, Daniel, “Finite groups with Sylow 2-subgroups of class two. II”, Trans. Amer. Math. Soc. 207 (1975), 103–126.CrossRefGoogle Scholar
[7]Glauberman, George, “Factorizations for 2-constrained groups”, preprint.Google Scholar
[8]Glauberman, George, “A characterization of the Goldschmidt groups”, (unpublished).Google Scholar
[9]Gorenstein, Daniel, Finite groups (Harper and Row, New York, Evanston, London, 1968).Google Scholar
[10]Gorenstein, Daniel and Harada, Koichiro, “A characterization of Janko's two new simple groups”, J. Fac. Sci. Univ. Tokyo Sect. I 16 (1969), 331–406 (1970).Google Scholar
[11]Gorenstein, Daniel and Harada, Koichiro, Finite groups whose 2-subgroups are generated by at most 4 elements (Memoirs Amer. Math. Soc., 147. Amer. Math. Soc., Providence, Rhode Island, 1974).CrossRefGoogle Scholar
[12]Gorenstein, Daniel and Lyons, Richard, “Nonsolvable finite groups with solvable 2-local subgroups”, J. Algebra 38 (1976), 453–522.CrossRefGoogle Scholar
[13]Gorenstein, Daniel and Walter, John H., “The characterization of finite groups with dihedral Sylow 2-subgroups. I”, J. Algebra 2 (1965), 85–151.CrossRefGoogle Scholar
[14]Gorenstein, Daniel and Walter, John H., “The characterization of finite groups with dihedral Sylow 2-subgroups -II”, J. Algebra 2 (1965), 218–270.CrossRefGoogle Scholar
[15]Gorenstein, Daniel and Walter, John H., “The characterization of finite groups with dihedral Sylow 2-subgroups. III”, J. Algebra 2 (1965), 354–393.CrossRefGoogle Scholar
[16]Gorenstein, Daniel and Walter, John H., “Centralizers of involutions in balanced groups”, J. Algebra 20 (1972), 284–319.CrossRefGoogle Scholar
[17]Huppert, B., Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[18]Martineau, R. Patrick, “On simple groups of order prime to 3”, Proc London Math. Soc. (3) 25 (1972), 213–252.CrossRefGoogle Scholar
[19]Sims, Charles C., “Computational methods in the study of permutation groups”, Computational problems in abstract algebra, 169–183 (Proc. Conf. Oxford,1967.Pergamon Press,Oxford, London, Edinburgh, New York, Toronto, Sydney, Paris, Braunschweig, 1970).CrossRefGoogle Scholar
[20]Stewart, W.B., “Groups having strongly self-centralizing 3-centralizers”, Proc. London Math. Soc. (3) 26 (1973), 653–680.CrossRefGoogle Scholar
[21]Suzuki, Michio, “A characterization of simple groups LF (2, p)”, J. Fac. Sci. Univ. Tokyo Sect. I 6 (1951), 259–293.Google Scholar
[22]Walter, John H., “The characterization of finite groups with abelian Sylow 2-subgroups”, Ann. of Math. (2) 89 (1969), 405–514.CrossRefGoogle Scholar