Homography
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 PGL(n + 1, F) when acting on a projective space of dimension n over a field F. Above definition of homographies shows that PGL(n + 1, F) may be identified to the quotient group GL(n + 1, F) / F I, where GL(n + 1, F) 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 (n + 1) × (n + 1).

When F is a Galois field GF(q) then the homography group is written PGL(n, q). For example, PGL(2, 7) acts on the eight points in the projective line over the finite field GF(7), while PGL(2, 4), which is isomorphic to the alternating group A, is the homography group of the projective line with five points.

The homography group PGL(n + 1, F) is a subgroup of the collineation group PΓL(n + 1, F) of the collineations of a projective space of dimension n. When the points and lines of the projective space are viewed as a block design, whose blocks are the sets of points contained in a line, it is common to call the collineation group the automorphism group of the design.
oulipo:avoid_letter[t], startend:end_checker, change_case:english_capital