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On a theorem of Wielandt concerning simply primitive groups

  • G. A. Jones (a1) and K. D. Soomro (a2)

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.

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References

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(1)Burnside, W.On some properties of groups of odd order. Proc. London Math. Soc. 33 (1901), 162185.
(2)Burnside, W.Theory of groups of finite order, 2nd. ed. (Cambridge University Press, London, 1911; reprinted Dover, New York, 1955).
(3)Chillag, D.On doubly transitive permutation groups of degree prime squared plus one. J. Austral. Math. Soc. (A) 23 (1977), 202206.
(4)Huppert, B.Endliche Gruppen I (Springer, Berlin, 1967).
(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), 5376.
(6)Jordan, C.Sur la lamite de transitivité des groupes non alternés. Bull. Soc. Math. Frnnce. 1 (1873), 4071.
(7)Sims, C. C.Graphs and finite permutation groups. Math. Z. 95 (1967), 7686.
(8)Soomro, K. D. Ph.D. thesis, University of Southampton, 1981.
(9)Wielandt, H.Finite permutation groups (Academic Press, New York, 1964).
(10)Wielandt, H.Permutation groups through invariant relations and invariant fimctions (Lecture Notes, Ohio State University, 1969).

On a theorem of Wielandt concerning simply primitive groups

  • G. A. Jones (a1) and K. D. Soomro (a2)

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