We will assume throughout this section that **P** is a Desarguesian projective space, so there exists a division ring F and vector space V over F so that **P** = **P**(V). Let **H** be a hyperplane of **P** and **A** = **P**\**H** be the affine space determined by **H**. Also, let O be a fixed point of **A**. It will be convenient to work in the affine space **A**. Recall that the points of **A** can be identified with the vectors in V and that the lines of **A** are the cosets of the 1-dimensional subspaces of V.

Let T = T(H) be the translation group of **A**, i.e., the group of all elations of **P** with axis **H**. Let G be the group of all collineations of **A**, and let G_{O} be the subgroup of G consisting of all collineations of **A** which fix the point O. [The facts that G is a group and that G_{O} is a subgroup of it are easy exercises.]

**Lemma 3.5.1:** (a) T is a normal subgroup of G.

(b) Each g in G can be uniquely written as g = tg_{O} where t in T and g_{O} in G_{O}.

This lemma says that every collineation of A can be described as a product of a translation and an element of G_{O}. Thus, we need only describe all the elements of these two subgroups in order to describe all collineations.

**Lemma 3.5.2:** Let t in T be an arbitrary translation. Choose an arbitrary point P of **A** and define P' = t(P). Viewing P and P' as vectors of V, we can describe t by:

**Def:** Let V be a vector space over the division ring F, and let be an automorphism of F. A map g of V into itself is called a * semilinear* map with

g(av) = (a)g(v).

*Example:*

Recall that in a finite field GF(q), with q = p^{e}, the map x x^{p} is an automorphism of the field. Let V be a 3-dimensional vector space over GF(2^{e}). The map g: V V given by g (x,y,z) = (y^{2}, z^{2}, x^{2}) is a semilinear map with companion automorphism being the squaring automorphism. To see this, let v = (x,y,z) and w = (x',y',z'), then g(v) =(y^{2},z^{2},x^{2}) and g(w) = (y'^{2},z'^{2},x'^{2}). Now, v + w = (x+x', y+y', z+z') and we have,

g(av) = ((ay)

**Lemma 3.5.3:** Every element of G_{O} is an additive map, that is, s in G_{O} implies that s(v+w) = s(v) + s(w), for all v,w in V.

**Theorem 3.5.4 **: Every element of G_{O} is a semilinear map of the vector space V.

We now turn to the projective space. It is easy to see that each collineation of **A** induces a collineation of **P**, but it is remarkable that the converse statement holds and this is the hard part of the following:

**Theorem 3.5.8 :** [The Fundamental Theorem of Projective Geometry] If **P** is a Desarguesian projective space and V a vector space such that **P** = **P**(V) then is a collineation of **P** if and only if there is a bijective semilinear map of V which induces .

We can now refine our view of the group of collineations of the projective space **P**, and see what role the central collineations play in this group.

**Def**: A collineation of the projective space **P** = **P**(V) is called a * projective collineation* (also known as a

**Theorem 3.6.1** : Every central collineation is a projective collineation.

**Theorem 3.6.7** : Every projective collineation of P is a product of central collineations.

The group of collineations of P is denoted by ** PL(n,F)** or