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 |
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Ternary fields |
I.2 |
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I.3 |
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I.4 |
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I.6 |
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II.1 |
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Cartesian groups |
II.2 |
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III.1 |
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III.2 |
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IVa.1 |
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Quasifields |
IVa.2 | There exist a line a und two distinct point u,v on a
such that
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Nearfields |
IVa.3 | There exist a line a and an involutorial, fixed point free,
permutation t of a such that
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IVb.1 | Dual of IVa.1 |
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IVb.2 | Dual of IVa.2 |
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IVb.3 | Dual of IVa.3 |
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V | There exist a line a and a point z on a such that
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Semifields |
VII.1 |
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Alternative fields |
VII.2 |
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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.
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