# Groups of central-axial collineations

### Related pages

Homologies and elations of P(K,T)
The Lenz classification
The Lenz-Barlotti classification
Let be a projective plane, and write . Given a point and a line the set

has center z and axis a

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 .

is axial with axis a ,
is central with center z ,
has axis a and it's center on b .

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.