Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-04-30T20:42:23.039Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of simple groups with no doubly transitive representations

Published online by Cambridge University Press:  09 April 2009

R. J. Clarke
R. J. Clarke
Affiliation:
Department of Pure Mathematics, University of AdelaideSouth Australia
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 first counterexample to the conjecture that all non a belian simple groups have doubly transitive permutation representations was pointed out by Parker in (1954), where he showed that the unitary groups PSU(4,4) had no doubly transitive representations. In this paper we generalize Parker's result to give an infinite class of simple groups having no doubly transitive permutation repreaentations. In this paper we generalize Parker's result to give an infinite class of simple groups having no doubly transitive permutation representations. Specifically, we prove THEOREM 1. The projective symplectic group PSp(4, q) has no doubly transitive permutation representation for q >2.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 2 , March 1975 , pp. 129 - 145
Copyright
Copyright © Australian Mathematical Society 1975

References

Carter, R. W. (1965), ‘Simple groups and simple Lie algebras’, J. London Math. Soc. 40, 193–240.CrossRefGoogle Scholar
Curtis, C. W. (1966), ‘The Steinberg character of a finite group with a BN-pair’, J. Algebra 4, 433–441.CrossRefGoogle Scholar
Higman, D. G. (1964), ‘Finite permutation groups of rank 3’, Math. Zeitschr. 86, 145–156.Google Scholar
Higman, D. G. and McLaughlin, J. E. (1965), ‘Rank 3 subgroups of finite symplectic and unitary groups’, J. reine u. angew. Math. 218, 174–189.Google Scholar
Huppert, B. (1967), Endliche Gruppen I (Springer-Verlag, Berlin 1967).CrossRefGoogle Scholar
Lang, S. (1956), ‘Algebraic groups over finite fields’, Amer. Math. J. 78, 555–563.CrossRefGoogle Scholar
Parker, E. T., ‘A simple group having no multiply transitive representation’, Proc. Amer. Math. Soc. 5, 606–611.Google Scholar
Srinavasan, Bhama (1968), ‘The characters of the finite symplectic group Sp (4, q)’, Trans. Amer. Math. Soc. 131, 488–425.Google Scholar
Tits, J. (1964), ‘Algebraic and abstract simple groups’, Ann. of Math. 80, 313–339.Google Scholar
Witt, E. (1937), ‘Die 5-fach transitiven Gruppen von Mathieu’, Abh. Math. Sem. Univ. Hamburg 12, 256–264.Google Scholar