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In [[projective geometry]] and [[linear algebra]], the '''projective orthogonal group''' PO is the induced [[Group action (mathematics)|action]] of the [[orthogonal group]] of a [[quadratic space]] ''V'' = (''V'',''Q'')<ref group="note">A quadratic space is a [[vector space]] ''V'' together with a [[quadratic form]] ''Q''; the ''Q'' is dropped from notation when it is clear.</ref> on the associated [[projective space]] P(''V''). Explicitly, the projective orthogonal group is the [[quotient group]]
:PO(''V'') = O(''V'')/ZO(''V'') = O(''V'')/{±''I''}
where O(''V'') is the orthogonal group of (''V'') and ZO(''V'')={±''I''} is the subgroup of all orthogonal [[scalar transformation]]s of ''V'' – these consist of the identity and [[reflection through the origin]]. These scalars are quotiented out because they act [[trivial action|trivially]] on the projective space and they form the [[Kernel (algebra)|kernel]] of the action, and the notation "Z" is because the scalar transformations are the [[center of a group|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. Explicitly:
:PSO(''V'') = SO(''V'')/ZSO(''V'')
where SO(''V'') is the special orthogonal group over ''V'' and ZSO(''V'') is the subgroup of orthogonal scalar transformations with unit [[determinant]]. Here ZSO is the center of SO, and is trivial in odd dimension, while it equals {±1} in even dimension – this odd/even distinction occurs throughout the structure of the orthogonal groups. By analogy with GL/SL and GO/SO, the projective orthogonal group is also sometimes called the '''projective ''general'' orthogonal group''' and denoted PGO.

Like the orthogonal group, the projective orthogonal group can be defined over any field and with varied quadratic forms, though, as with the ordinary orthogonal group, the main emphasis is on the ''real'' ''positive definite'' projective orthogonal group; other fields are elaborated in [[#Generalizations|generalizations]], below. Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups.

Like the [[spin group]]s and [[pin group]]s, which are covers rather than quotients of the (special) orthogonal groups, the projective (special) orthogonal groups are of interest for (projective) geometric analogs of Euclidean geometry, as related [[Lie group]]s, and in [[representation theory]].

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).

== Structure ==

=== Odd and even dimensions ===
[[File:SO-O-PSO-PO-2k.svg|right]]

The structure of PO differs significantly between odd and even dimension, fundamentally because in even dimension, [[reflection through the origin]] is orientation-preserving, while in odd dimension it is orientation-reversing (<math>-I \in \operatorname{SO}(2k)</math> but <math>-I \not\in \operatorname{SO}(2k+1)</math>). This is seen in the fact that each odd-dimensional real projective space is orientable, while each even-dimensional real projective space of positive dimension is non-orientable. At a more abstract level, the [[Lie algebra]]s of odd- and even-dimensional projective orthogonal groups form two different families: <math>B_k = \mathfrak{so}_{2k+1}, D_k = \mathfrak{so}_{2k}.</math>

Thus, O(2''k''+1) = SO(2''k''+1) × {±''I''},<ref group="note">This product is an [[internal direct sum]] – a product of subgroups – not just an abstract [[external direct sum]].</ref>
while <math>\operatorname{O}(2k) \neq \operatorname{SO}(2k) \times \{\pm I\}</math> and is instead a non-trivial [[Group extension#Central extension|central extension]] of PO(2''k'').

Beware that PO(2''k''+1) is isometries of '''R'''P<sup>2''k''</sup> = P('''R'''<sup>2''k''+1</sup>), while PO(2''k'') is isometries of '''R'''P<sup>2''k''−1</sup> = P('''R'''<sup>2''k''</sup>) – the odd-dimensional (vector) group is isometries of even-dimensional projective space, while the even-dimensional (vector) group is isometries of odd-dimensional projective space.

[[File:SO-O-PSO-PO-2k+1.svg|right]]
In odd dimension, <math>\operatorname{SO}(2k+1) \cong \operatorname{PSO}(2k+1) = \operatorname{PO}(2k+1),</math><ref group="note">The [[isomorphism]]/equality distinction in this equation is because the context is the 2-to-1 quotient map O → PO – PSO(2''k''+1) and PO(2''k''+1) are equal subsets of the target (namely, the whole space), hence the equality, while the induced map SO → PSO is an isomorphism but the two groups are subsets of different spaces, hence the isomorphism rather than an equality. See {{Harv | Conway | Smith | 2003 | loc = [https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA34 p. 34]}} for an example of this distinction being made.</ref> so the group of projective isometries can be identified with the group of rotational isometries.

In even dimension, SO(2''k'')&nbsp;→&nbsp;PSO(2''k'') and O(2''k'')&nbsp;→&nbsp;PO(2''k'') are both 2-to-1 covers, and PSO(2''k'') < PO(2''k'') is an [[Index of a subgroup|index]] 2 subgroup.

=== General properties ===
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. This is analogous to SO being maximal compact in SL and O being maximal compact in GL.

== Representation theory ==
{{Further|Projective representation}}
PO is of basic interest in representation theory: a group homomorphism ''G''&nbsp;→&nbsp;PGL is called a '''[[projective representation]]''' of ''G,'' just as a map ''G''&nbsp;→&nbsp;GL is called a '''linear representation''' of ''G'', and just as any linear representation can be reduced to a map ''G''&nbsp;→&nbsp;O (by taking an invariant inner product), any projective representation can be reduced to a map ''G''&nbsp;→&nbsp;PO.

See [[projective linear group#Representation theory|projective linear group: representation theory]] for further discussion.

== Subgroups ==
{{See also|Spin group#Discrete subgroups|Binary polyhedral group}}
Subgroups of the projective orthogonal group correspond to subgroups of the orthogonal group that contain −''I'' (that have [[central symmetry]]). 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 polytope]]s – these correspond to the (discrete) point groups that include central symmetry. Compare with [[Spin group#Discrete subgroups|discrete subgroups of the Spin group]], particularly the 3-dimensional case of [[binary polyhedral group]]s.

For example, in 3 dimensions, 4 of the 5 [[Platonic solid]]s have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the [[stellated octahedron]] has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.

== Topology ==
{{See also|Orthogonal group#Covering and quotient groups}}
PO and PSO, as centerless topological groups, are at the bottom of a sequence of [[covering group]]s, whose top are the ([[simply connected]]) [[Pin group]]s or [[Spin group]], respectively:
:Pin<sub>±</sub>(''n'')&nbsp;→&nbsp;O(''n'')&nbsp;→&nbsp;PO(''n'').
:Spin(''n'')&nbsp;→&nbsp;SO(''n'')&nbsp;→&nbsp;PSO(''n'').
These groups are all [[compact real form]]s of the same Lie algebra.

These are all 2-to-1 covers, except for SO(2''k''+1)&nbsp;→&nbsp;PSO(2''k''+1) which is 1-to-1 (an isomorphism).

=== Homotopy groups ===
[[Homotopy group]]s above <math>\pi_1</math> do not change under covers, so they agree with those of the orthogonal group. The lower homotopy groups are given as follows.
:<math>\pi_0(\operatorname{PSO}) \cong 1</math>
:<math>\pi_0(\operatorname{PO}(2k)) \cong \mathbf{Z}/2\mathbf{Z}, \pi_0(\operatorname{PO}(2k+1)) \cong 1.</math>

The fundamental group of (centerless) PSO(''n'') equals the center of (simply connected) Spin(''n''), which is always true about covering groups:
:<math>\pi_1(\operatorname{PSO}(n)) = \pi_1 (\operatorname{PO}(n)) = \operatorname{Z}(\operatorname{Spin}(n)).</math>
Using the [[Spin group#Center|table of centers of Spin groups]] yields (for <math>k \geq 1</math>):
:<math>\pi_1 (\operatorname{PSO}(4k)) = \mathbf{Z}/2\mathbf{Z} \oplus \mathbf{Z}/2\mathbf{Z},</math>
:<math>\pi_1 (\operatorname{PSO}(4k+2)) = \mathbf{Z}/4\mathbf{Z},</math>
:<math>\pi_1 (\operatorname{PSO}(2k+1)) = \pi_1(\operatorname{SO}(2k+1)) = \mathbf{Z}/2\mathbf{Z},</math>
In low dimensions:
:<math>\pi_1 (\operatorname{PSO}(1)) = 1,</math> as the group is trivial.
:<math>\pi_1 (\operatorname{PSO}(2)) = \mathbf{Z},</math> as it is topologically a circle, though note that the preimage of the identity in Spin(2) is <math>\mathbf{Z}/4\mathbf{Z},</math> as for other <math>4k+2.</math>

=== Homology groups ===
{{Empty section|date=May 2010}}

=== Bundles ===
{{Expand section|date=May 2010}}
Just as the orthogonal group is the [[structure group]] of [[vector bundle]]s, the projective orthogonal group is the structure group of [[projective bundle]]s, and the corresponding [[classifying space]] is denoted BPO.

== Generalizations ==
As with the orthogonal group, the projective orthogonal group can be generalized in two main ways: changing the field or changing the quadratic form. Other than the real numbers, primary interest is in complex numbers or finite fields, while (over the reals) quadratic forms can also be [[indefinite form]]s, 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, {{nowrap|1=''I'' = −''I''}}, and the determinant needs to be replaced by the [[Orthogonal group|Dickson invariant]].

=== Finite fields ===
{{Expand section|date=May 2010}}
The projective orthogonal group over a finite field is used in the construction of a family of finite [[simple group]]s of [[Group of Lie type|Lie type]], namely the [[Chevalley group]]s of type D<sub>''n''</sub>. 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).<ref>[[ATLAS of Finite Groups|ATLAS]], [https://books.google.com/books?id=38fEMl2-Fp8C&dq=%22projective%20special%20orthogonal%20group%22&pg=PR11 p. xi]</ref> The quasideterminant can be defined as (−1)<sup>''D''</sup>, where ''D'' is the [[Orthogonal group|Dickson invariant]] (it is the determinant defined by the Dickson invariant), or in terms of the dimension of the fixed space.

== Notes ==
{{Reflist| group = note}}

== See also ==
* [[Projective linear group]]
* [[Projective unitary group]]
* [[Orthogonal group]]
* [[Spin group]]

== References ==
{{Refimprove|date=May 2010}}
{{Reflist}}
{{refbegin}}
* {{Citation
| publisher = A K Peters, Ltd.
| isbn = 978-1-56881-134-5
| last1 = Conway
| first1 = John Horton
| authorlink1 = John Horton Conway
| first2 = Derek Alan | last2 = Smith
| title = On quaternions and octonions
| date = 2003-02-07
| chapter = 3.7 The Projective or Elliptic Groups
| pages = [https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA34 34]
}}
* Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GO<sub>''n''</sub>(''q''), SO<sub>''n''</sub>(''q''), PGO<sub>''n''</sub>(''q''), and PSO<sub>''n''</sub>(''q''), and O<sub>''n''</sub>(''q'')." §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi–xii, 1985.
{{refend}}

==External links==
* {{MathWorld |title=Projective General Orthogonal Group |urlname=ProjectiveGeneralOrthogonalGroup}}
* {{MathWorld |title=Projective Special Orthogonal Group |urlname=ProjectiveSpecialOrthogonalGroup}}

{{DEFAULTSORT:Projective Orthogonal Group}}
[[Category:Lie groups]]
[[Category:Projective geometry]]
[[Category:Quadratic forms]]