Related pagesTranslation planes
Sharply 2-transitive groups
Theorem. Let K be a nearfield. Then the translation plane P(K) is - transitive for each line l through (0) and (l,(0))-transitive for each line l through
Nearfields are characterized by this property.
Theorem. Let (K,T) be a ternary field. Then the projective plane P(K,T) is - transitive and - transitive if and only if (K,T) is a nearfield.
In particular, the projective planes of Lenz-Barlotti type at least IVa.2 are exactly the planes P(K) over nearfields K. The finite nearfields are completely classified.