# Homologies and elations of P(K,T)

Let (K,T) be a ternary field. Let be the projective plane over (K,T), and write . We are going to describe the groups of central-axial collineations, for some values of a and z.

First, let the axis a be the line at infinity.

Next, we will describe some groups with the vertical line [0] as axis.
• where a-1 is defined by the equation aa-1=1.

Moreover,

All the groups described above fix the line at infinity. Unfortunately, sometimes you need to know central-axial collineations moving the line at infinity. We will compute one such group, using [0] as axis and (0,0) as center.

Let a be an element of K, distinct from 0. Define a' by a+a'=0. For each t in K, there exists exactly one element

The map is a bijection with Moreover, if x is an element of K distinct from a', there exists exactly one

and the map

is a bijection. Using these notations, we define maps defined on the sets of points and lines of P(K,T), respectively.

This map will be called Finally, define

We get

Contributed by Hauke Klein
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