# The Lenz-Barlotti classification

The Lenz-Barlotti classification refines the Lenz classification of projective planes by considering transitive groups of homologies. Let be a projective plane, and write . The Lenz-Barlotti figure of is defined to be the following set

is (a,z)-transitive .

Obviously, the Lenz-Barlotti figure is invariant under the action of , and

is the Lenz figure of . The Lenz-Barlotti classification lists all possible structure types for the Lenz-Barlotti figure of a projective plane. Each of the Lenz types will be divided in one or more Lenz-Barlotti types.

Theorem. Let be a projective plane. Then exactly one of the following statements is true.

Type Lenz-Barlotti figure Coordinatizing ternary field
I.1 Ternary fields
I.2
I.3
I.4 with three non collinear points u,v,w.
I.6 where and is bijective.
II.1 Cartesian groups
II.2
III.1 where .
III.2 where .
IVa.1 for a line a. Quasifields
IVa.2 There exist a line a und two distinct point u,v on a such that Nearfields
IVa.3 There exist a line a and an involutorial, fixed point free, permutation t of a such that
IVb.1 Dual of IVa.1
IVb.2 Dual of IVa.2
IVb.3 Dual of IVa.3
V There exist a line a and a point z on a such that Semifields
VII.1 Alternative fields
VII.2 Skewfields

The original list of all Lenz-Barlotti types was slightly longer, but, as it turned out, some types does not exist. These missing types are I.5, I.7, I.8 and II.3. Whether there exists a projective plane of Lenz-Barlotti type I.6 seems to be an open problem. The Lenz-Barlotti type IVa.3, and its dual IVb.3, is somewhat exceptional. In fact, there is exactly one plane of this type.

Contributed by Hauke Klein
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