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Translation planes
Finite spreads
Let V be a non trivial vector space over a skewfield K. A spreadset in V is a set of endomorphisms of V such that the following three conditions hold.

Of course, if the dimension of V is finite, the set is a spreadset if and only if and given any two vectors v,w in V, v distinct from zero, there is exactly one such that Av=w.

Spreadsets are almost the same objects as quasifields. In fact, the following simple theorem holds.

Theorem. Let K be a quasifield. Then K becomes a vector space over its kernel Ker(K) and

is a spreadset in this vector space. Here, la denotes the left multiplication by the element a, i.e. la(x)=ax.

Conversely, given a spreadset in a vector space V, we choose an element e in V distinct from zero, and for each v in V there exists exactly one with . Using this notation, we define a multiplication on V by

Then V becomes a quasifield and the original spreadset is the set of left multiplications in this quasifield.

Note that the spreadset is an additive subgroup of End(V) if and only if K is a semifield. Moreover, is a subgroup of GL(V) if and only if K is a nearfield.

Using the construction in the theorem, the kernel of a quasifield becomes the centralizer of the corresponding spreadset, i.e.

Each spreadset yields a spread, and we get a translation plane, which will be written as .

Theorem. Given a spreadset in a vector space V, the translation plane is desarguesian if and only if

See also

Contributed by Hauke Klein
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