Near nearfield planes
If \((F,+,\cdot)\) is a planar nearfield then we know that the affine plane over \(F\) admits the two groups \(G_ 1,G_ 2\) of affine perspectivities: \(G_ 1\) (resp. \(G_ 2)\) consists of all homologies with the \(x\)-axis \(X\) \((y\)-axis \(Y)\) as axis and the \(y\)-axis \(Y\) \((x\)-axis \(X)\) as direction and these two groups are acting transitively on the set \(Y\backslash X\) (resp. \(X\backslash Y)\). Both groups, \(G_ 1\) and \(G_ 2\) are isomorphic to the multiplicative group \((F^*,\cdot)\) of the nearfield. The authors call a translation plane \(\pi\) of finite odd order \(k\) a near nearfield plane if this plane admits at least two affine homology groups \(H_ 1\) and \(H_ 2\) each of order \((k-1)/2\). By this definition any affine plane over a finite nearfield \(F\) belongs to this class if \((F^*,\cdot)\) contains a subgroup of index 2. This is the case for all finite Dickson nearfields (including fields) of odd order and for the exceptional nearfields \(F\) of type (III) \((| F|=7^ 2)\) and type (IV) \((| F|=23^ 2)\) [cf. \textit{H. Wähling}, Theorem der Fastkörper (1987; Zbl 0669.12014), p. 373]. The authors show firstly that each near nearfield plane \(\pi\) admits always such homology groups \(H_ 1\) and \(H_ 2\) where the axis of \(H_ 2\) is the direction of \(H_ 1\) and vice versa. Then they show that, besides these mentioned nearfield examples, the only near nearfield planes are the exceptional Lüneburg plane of order \(7^ 2\) and the generalized André planes [for def. cf. \textit{H. Lüneburg}, Translation planes (1980; Zbl 0446.51003)].
scientific article; zbMATH DE number 64285