An oval Ω in a finite projective plane is said to be 2-transitive if the plane admits a collineation group G fixing Ω and acting 2-transitively on its points. In the order n of the plane is assumed to be even then the following result is proved.Theorem. If G fixes an external line and acts 2-transitively on Ω then either n ∈ {2, 4} or n ≡ 0 mod 8 and the Sylow 2-subgroups of G are generalized quaternion groups.The result is obtained by examining the action of G on a G-invariant family of pairwise disjoint ovals (including Ω) with a common knot.
On two-transitive ovals in projective planes of even order / Bonisoli, Arrigo; G., Korchmaros. - In: ARCHIV DER MATHEMATIK. - ISSN 0003-889X. - STAMPA. - 65:(1995), pp. 89-93.
On two-transitive ovals in projective planes of even order
BONISOLI, Arrigo;
1995
Abstract
An oval Ω in a finite projective plane is said to be 2-transitive if the plane admits a collineation group G fixing Ω and acting 2-transitively on its points. In the order n of the plane is assumed to be even then the following result is proved.Theorem. If G fixes an external line and acts 2-transitively on Ω then either n ∈ {2, 4} or n ≡ 0 mod 8 and the Sylow 2-subgroups of G are generalized quaternion groups.The result is obtained by examining the action of G on a G-invariant family of pairwise disjoint ovals (including Ω) with a common knot.
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