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Subplanes of finite projective planes
Definition. A proper subplane of a projective plane is called a Baer subplane if the following two conditions are true.

Of course, Baer subplanes may be characterized in terms of coordinatizing ternary fields.

Lemma. Let be a subplane of a projective plane . Let (o,e,u,v) be a quadrangle contained in the given subplane and let (K,T) be the corresponding ternary field. Then is a subfield of (K,T). The subplane is a Baer subplane of if and only if the following three conditions hold.

If is the projective plane over a vector space, then there is a more convenient description of Baer subplanes.

Lemma. Let V be a three-dimensional vector space over a skewfield K. Let be a subplane of PG(V). Then there exist a subskewfield P of K and a basis e1,e2,e3 of V such that The subplane is a Baer subplane if and only if K is of dimension 2 over P as a left vector space and as a right vector space, respectively.


Contributed by Hauke Klein
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