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On a theorem of Wielandt concerning simply primitive groups

Published online by Cambridge University Press:  24 October 2008

G. A. Jones  and
K. D. Soomro
G. A. Jones
Affiliation:
University of Southampton, Pakistan
K. D. Soomro
Affiliation:
University of SindPakistan
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Extract

Let G be a simply primitive permutation group on a set Ω of order p2, where p is a prime (necessarily odd). In theorem 27·2 of (9), Wielandt states without proof:

Theorem A. (i) ¦G¦ is not divisible by p3;

(ii) if G has a pair of Sylow p-subgroups with nontrivial intersection, then G has an imprimitive subgroup of index 2 which is the direct product of two intransitive groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 419 - 423
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Burnside, W.On some properties of groups of odd order. Proc. London Math. Soc. 33 (1901), 162–185.Google Scholar
(2)Burnside, W.Theory of groups of finite order, 2nd. ed. (Cambridge University Press, London, 1911; reprinted Dover, New York, 1955).Google Scholar
(3)Chillag, D.On doubly transitive permutation groups of degree prime squared plus one. J. Austral. Math. Soc. (A) 23 (1977), 202–206.CrossRefGoogle Scholar
(4)Huppert, B.Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
(5)Jones, G. A.Abelian subgroups of simply primitive groups of degree p3, where p is prime. Quart. J. Math. Oxford Ser. 2, 30 (1979), 53–76.CrossRefGoogle Scholar
(6)Jordan, C.Sur la lamite de transitivité des groupes non alternés. Bull. Soc. Math. Frnnce. 1 (1873), 40–71.CrossRefGoogle Scholar
(7)Sims, C. C.Graphs and finite permutation groups. Math. Z. 95 (1967), 76–86.CrossRefGoogle Scholar
(8)Soomro, K. D. Ph.D. thesis, University of Southampton, 1981.Google Scholar
(9)Wielandt, H.Finite permutation groups (Academic Press, New York, 1964).Google Scholar
(10)Wielandt, H.Permutation groups through invariant relations and invariant fimctions (Lecture Notes, Ohio State University, 1969).Google Scholar