the three following theorems. Given two projective frames of a projective space P, there is exactly one homography of P that maps the first frame onto the second one. If the dimension of a projective space P is at least two, every collineation of P is the composition of an automorphic collineation and a homography. In particular, over the reals, every collineation of a projective space of dimension at least two is a homography. Every homography is the composition of a finite number of perspectivities. In particular, if the dimension of the implied projective space is at least two, every homography is the composition of a finite number of central collineations. If projective spaces are defined by means of axioms (synthetic geometry), the third part is simply a definition. On the other hand, if projective spaces are defined by means of linear algebra, the first part is an easy corollary of the definitions. Therefore, the proof of the first part in synthetic geometry, and the proof of the third part in terms of linear algebra both are fundamental steps of the proof of the equivalence of the two ways of defining projective spaces. Homography groups As every homography has an inverse mapping and the composition of two homographies is another, the homographies of a given projective space form a group. For example, the Möbius group is the homography group of any complex projective line. As all the projective spaces of the same dimension over the same field are isomorphic, the same is true for their homography groups. They are therefore considered as a single group acting on several spaces, and only the dimension and the field appear in the notation, not the specific projective space. Homography groups also called projective linear groups are denoted when acting on a projective space of dimension n over a field F. Above definition of homographies shows that may be identified to the quotient group , where is the general linear group of the invertible matrices, and F×I is the group of the products by a nonzero element of F of the identity matrix of size . When F is a Galois field GF(q) then the homography group is written . For example, acts on the eight points in the projective line over the finite field GF(7), while , which is isomorphic to the alternating group A5, is the homography group of the projective line