is (a,z)-transitive
.
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.
|
|
|