Projective orthogonal group

These scalars are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" is because the scalar transformations are the center of the orthogonal group.

The projective special orthogonal group, PSO, is defined analogously, as the induced action of the special orthogonal group on the associated projective space.

Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups.

More intrinsically, the (real positive definite) projective orthogonal group PO can be defined as the isometries of elliptic space (in the sense of elliptic geometry), while PSO can be defined as the orientation-preserving isometries of elliptic space (when the space is orientable; otherwise PSO = PO).

At a more abstract level, the Lie algebras of odd- and even-dimensional projective orthogonal groups form two different families:

PSO and PO are centerless, as with PSL and PGL; this is because scalar matrices are not only the center of SO and O, but also the hypercenter (quotient by the center does not always yield a centerless group).

PSO is the maximal compact subgroup in the projective special linear group PSL, while PO is maximal compact in the projective general linear group PGL.

See projective linear group: representation theory for further discussion.

As always with a quotient map (by the lattice theorem), there is a Galois connection between subgroups of O and PO, where the adjunction on O (given by taking the image in PO and then the preimage in O) simply adds −I if absent.

Of particular interest are discrete subgroups, which can be realized as symmetries of projective polytopes – these correspond to the (discrete) point groups that include central symmetry.

Other than the real numbers, primary interest is in complex numbers or finite fields, while (over the reals) quadratic forms can also be indefinite forms, and are denoted PO(p,q) by their signature.

The complex projective orthogonal group, PO(n,C) should not be confused with the projective unitary group, PU(n): PO preserves a symmetric form, while PU preserves a hermitian form – PU is the symmetries of complex projective space (preserving the Fubini–Study metric).

In fields of characteristic 2 there are added complications: quadratic forms and symmetric bilinear forms are no longer equivalent, I = −I, and the determinant needs to be replaced by the Dickson invariant.

The orthogonal group over a finite field, O(n,q) is not simple, since it has SO as a subgroup and a non-trivial center ({±I}) (hence PO as quotient).

These are both fixed by passing to PSO, but PSO itself is not in general simple, and instead one must use a subgroup (which may be of index 1 or 2), defined by the spinor norm (in odd characteristic) or the quasideterminant (in even characteristic).