## 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 .

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.

Version