The projective general orthogonal group 
is the quotient group
of the general orthogonal group 
by the scalar matrices
that it contains,
 |
Here 
denotes the group center of a group 
,
and 
is a nonsingular quadratic form on 
. This quotient group
is the projective image of 
and has a faithful
group action on the associated quadric in projective
space. For odd 
, the scalar subgroup being factored
out is 
, where 
is the 
identity matrix.
The image of the special orthogonal group is the projective
special orthogonal group. As with the nonprojective groups,
the notation may include a superscript 
or 
in even dimension to specify
the type of the quadratic form.
, 
, 
, and 
, and 
." §2.4 in