Projective group

From Encyclopedia of Mathematics
Revision as of 17:27, 7 February 2011 by (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

in variables over a skew-field

The group of transformations of the -dimensional projective space induced by the linear transformations of . There is a natural epimorphism

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


[1] J.A. Dieudonné, "La géométrie des groupes classiques" , Springer (1955)


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
How to Cite This Entry:
Projective group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article