Projective Special Orthogonal Group

The projective special orthogonal group is the quotient group of the special orthogonal group by the scalar matrices it contains,

Here denotes the group center of a group . It is the projective image of and is a normal subgroup of the projective general orthogonal group. In even dimension, a superscript or records the type of the nonsingular quadratic form .

Unlike the projective special linear group and most projective special unitary groups, is not in general a simple group. The finite simple groups in the orthogonal families instead arise, apart from exceptions in low dimension, as projective images of appropriate commutator subgroups.