The Lenz classification

Related pages

The Lenz-Barlotti classification
Let be a projective plane, and write . The Lenz figure of is defined to be the following set of flags

and is (a,z)-transitive

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
I Ternary fields
II Cartesian groups
III There exist a point z and a line l with such that

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
VII Alternative fields

The Lenz types IVa,V and VII are translation planes. In terms of translation axes, we get the following table.

Lenz type Interpretation
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.
VII Moufang planes

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.

Contributed by Hauke Klein
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