The fundamental theorem of projective geometry

Let V be a three-dimensional vector space over a skewfield K. Given a semilinear map we get induced maps

defined on the sets Ti(V) of i-dimensional subspaces of V. Obviously, these induced maps yield an automorphism of the projective plane PG(V). We define

Theorem. The group is the full automorphism group of the projective plane PG(V).

The subgroup

is an invariant subgroup of , called the projective group of PG(V). It equals the subgroup generated by all central-axial collineations of PG(V). Moreover,

is isomorphic to the group Out(K) of outer automorphisms of the skewfield K. In particular, we have if and only if each automorphism of K is an inner automorphism.

Contributed by Hauke Klein
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