Related pagesThe Lenz-Barlotti classification
Obviously, the Lenz figure is invariant under the action of . The Lenz classification of projective planes states that there are only 7 possible types of Lenz figures.
Theorem. Let be a projective plane. Then exactly one of the following seven statements is true.
|Lenz type||Lenz figure||Coordinatizing ternary field|
|III||There exist a point z and a line l with
|Special Cartesian groups|
|IVa||There exists a line a such that||Quasifields|
|IVb||There exists a point z such that||Dual of IVa|
|V||There exist a line a and a point z on a such that||Semifields|
The Lenz types IVa,V and VII are translation planes. In terms of translation axes, we get the following table.
|IVa||Translation planes with a unique translation axis.|
|IVb||Dual translation planes with unique translation center|
|V||The plane is a translation plane and a dual translation plane.|
In particular, we have the theorem of Skornjakov and San-Soucie.
Theorem. A projective plane which admits two distinct translation axes is a Moufang plane.