Subplanes of PG(V)

Related pages

Subplanes of P(K,T)
Baer subplanes
Let V be a three-dimensional vector space over a skewfield L. We have the projective plane over V, as usual denoted by PG(V). Let K be a subskewfield of L, and let e1, e2,e3 be a basis of V over L. Then W=e1K+e2K+e3K becomes a three-dimensional vector space over K.

Given any subspace U of W over K, then the span <U> over L is a subspace of V over L with dimKU=dimL<U>, and In this sense, we may think of one- or two-dimensional K-subspaces of W as special one- or two-dimensional L-subspaces of V, respectively. Then the projective plane PG(W) becomes a subplane of PG(V).

In fact, all subplanes of PG(V) arise in this way.

Contributed by Hauke Klein
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