The projective general linear group 
over a field 
is the quotient group of
the general linear group by its group
center,
 |
(1)
|
Here 
is the 
identity matrix. Two invertible matrices
therefore determine the same element of 
precisely when one is a nonzero scalar
multiple of the other. The quotient group has a
faithful group action on the projective
space 
because the group action of scalar
matrices is trivial. In particular, the group action
of 
on the projective line is by linear
fractional transformations.
For the finite field 
, writing 
for the group order of a
finite group 
gives
 |
(2)
|
The projective special linear group is a normal subgroup, and
 |
(3)
|
so its index is 
.
, 
, 
, and 
." §2.1 in