On an Oval with the Four Point Pascalian Property

In this paper it is proved that a finite translation plane of order n ≡ 3 (mod 4) which contains an oval with the four point Pascalian property, or a finite dual translation plane of order n ≡ 3 (mod 4) which contains an oval with the four point Pascalian property, can be coordinatized by a commutative semifield.

Translation planes of even order in which the dimension has only one odd factor

LetGbe an irreducible subgroup of the linear translation complement of a finite translation plane of orderqdwhereqis a power of2.GF(q)is in the kernel andd=2srwhereris an odd prime. A prime factor of|G|must divide(qd+1)∏i=1d(qi−1).One possibility (there are no known examples) is thatGhas a normal subgroupWwhich is aW-group for some primeW.The maximal normal subgroup0(G)satisfies one of the following:1. Cyclic. 2. Normal cyclic subgroup of indexrand the nonfixed-point-free elements in0(G)have orderr. 3.0(G)contains a groupWas above.

Indicator Sets in an Affine Space of any Dimension

It is well known that a translation plane can be represented in a vector space over a field F where F is a subfield of the kernel of a quasifield which coordinatizes the plane [1; 2; 4, p.220; 10]. If II is a finite translation plane of order qr (q = pn, p any prime), then II may be represented in V2r(q), the vector space of dimension 2r over GF(q), as follows:(i) The points of II are the vectors in V = V2r(q)(ii) The lines of II are(a) A set of qr + 1 mutually disjoint r-dimensional subspaces of V.(b) All translates of in V.(iii) Incidence is inclusion.