Related pagesHomologies and elations of P(K,T)
The Lenz classification
The Lenz-Barlotti classification
is a subgroup of acting freely on for each line l through the center z.
Definition. The projective plane is said to be (a,z)-transitive if the group is transitive on for each line l through z. As it turns out, the plane is (a,z)-transitive if and only if it is (a,z)-Desarguesian.
In fact, each of the following sets is a subgroup of .
where b is another line of .
In particular, is a group, called the group of translations with axis a. Moreover, we have the following important result.
Theorem. The group acts freely on . If there exist two distinct points u,v on a with , then the translation group is abelian.
Given any two lines a,b, not necessarily distinct, the group acts without fixed points on the set The plane is called (a,b)-transitive if this action is in fact sharply transitive. The line a is a translation axis if is (a,a)-transitive. Then is a so-called translation plane.