## Related pagesTranslations, contractions and shearsHomologies and elations of P(K,T) Groups of central-axial collineations Baer's lemma |

**Definition.**
is a central collineation with center
if
fixes all lines through the point z, i.e.
for each line
.

Dually, is an axial collineation with axis if fixes all points on the line a.

We have the following basic results.

**Lemma.** If
is a non-trivial collineation of
,
then
has at most one center and one axis, respectively.

We will write and to denote the center and axis of a non-trivial central or axial collineation , respectively.

**Lemma.** A collineation is central if and only if
the collineation is axial.

Consequently, these collineations are called central-axial collineations of . Moreover, if is a non-trivial central-axial collineation, then is a homology if Otherwise, is an elation if Finally, the identity is defined to be a homology and an elation.

**Lemma.** Let
be a central-axial collineation. If p is a point with
then the line
is fixed by
.
Dually, if l is a line with
then
is a fixed point of
.

**Lemma.** Let
be a non-trivial central-axial collineation of
.
Then

Version