# Projective group

*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

#### References

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

#### Comments

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].

#### References

[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: http://encyclopediaofmath.org/index.php?title=Projective_group&oldid=18829