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Transitive Extensions of a Class of Doubly Transitive Groups

Published online by Cambridge University Press:  22 January 2016

Michio Suzuki
Michio Suzuki*
Affiliation:
University of Illinois
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1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3 (q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 159 - 169
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Burnside, W., Theory of groups, 2nd ed. (1911).Google Scholar
[2] R.-Hughes, D., Combinatorial analysis; t-designs and permutation groups, Proc. Sympos. Pure Math., vol. VI (1962), 39–41.CrossRefGoogle Scholar
[3] Hughes, D. R., Extensions of designs and groups: The unitary groups, Math. Z. (to appear).Google Scholar
[4] Nagao, H., On multiply transitive permutation groups I, Nagoya J. of Math. 27 (1966), 15–19.CrossRefGoogle Scholar
[5] Ree, R., A family of simple groups associated with the simple Lie algebra of type (Gî), Amer. J. Math., 83 (1961), 432–462.CrossRefGoogle Scholar
[6] Suzuki, M., A characterization of simple groups LF(2, p) , Jour. Fac. Sci. Univ. of Tokyo, 1–6 (1951), 259–293.Google Scholar
[7] Suzuki, M., A new type of simple groups of finite order, Proc. Nat. Acad. Sci. USA, 46 (1960), 868–870.CrossRefGoogle ScholarPubMed
[8] Suzuki, M., On a class of doubly transitive groups II, Annals of Math. 79 (1964), 514–589.CrossRefGoogle Scholar
[9] Wielandt, H., Über den Transitivitätsgrad von Permutationsgruppen, Math. Z. 74 (1960), 297–298.CrossRefGoogle Scholar
[10] Witt, E., Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg Univ., 12 (1937), 256–264.CrossRefGoogle Scholar
[11] Zassenhaus, H., Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen, Abh. Math. Sem. Hamburg Univ. 11 (1936), 17–40.CrossRefGoogle Scholar