defined on the sets T_{i}(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.

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