Roughly speaking, it shows that all such groups are similar to the general linear group over a field.

For example, we have the general linear groups over finite fields.

Basic examples are , the general linear groups over the complex numbers.

For subgroups of the linear group, additional conditions must be imposed on the flags.

That is to say, the general linear group acts transitively on the set of all complete flags.

For general linear groups this was already known by the work of .

For the general linear group, we get as the Cartan involution.

The Lie group that it generates is the special linear group.

This is the affine analogue of the special linear group.

All matrix groups are subgroups of some general linear group.