in variables over a skew-field
with as kernel the group of homotheties (cf. Homothety) of , which is isomorphic to the multiplicative group of the centre of . The elements of , called projective transformations, are the collineations (cf. Collineation) of . Along with , which is also called the full projective group, one also considers the unimodular projective group , and, in general, groups of the form , where is a linear group.
For the group is simple, except for the two cases and or 3. If is the finite field of elements, then
|||J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1955) MR0072144 Zbl 0067.26104|
The groups for are the images of under . For a brief resumé on the orders of the other finite classical groups, like , and their simplicity cf. e.g. [a1].
|[a1]||R.W. Carter, "Simple groups of Lie type" , Wiley (Interscience) (1972) pp. Chapt. 1|
Projective group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_group&oldid=18829