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Translation planes
Lenz-Barlotti classification
Sharply 2-transitive groups
A nearfield is an associative quasifield. In more concrete terms, a nearfield consists of a set K and two binary operations such that the following axioms hold.

• (K,+) is a group. Let 0 be its neutral element.
• We have 0x=0 for all elements x of K.
• is a group.
• .
• .
Since each nearfield is a quasifield, we get a translation plane P(K) associated to a nearfield K. These translation planes admit transitive groups of homologies, in fact the following theorem holds.

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.