Projective orthogonal group

<h2 id="structure">Structure</h2>
<h3>Odd and even dimensions</h3>

The structure of PO differs significantly between odd and even dimension, fundamentally because in even dimension, <a href="/facts/Reflection_through_the_origin/UzswAm8N">reflection through the origin</a> is orientation-preserving, while in odd dimension it is orientation-reversing (
 
 
 
 −
 I
 ∈
 SO
 ⁡
 (
 2
 k
 )
 
 
 {\displaystyle -I\in \operatorname {SO} (2k)}
 
 but 
 
 
 
 −
 I
 ∉
 SO
 ⁡
 (
 2
 k
 +
 1
 )
 
 
 {\displaystyle -I\not \in \operatorname {SO} (2k+1)}
 
). 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 <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a> of odd- and even-dimensional projective orthogonal groups form two different families: 
 
 
 
 
 B
 
 k
 
 
 =
 
 
 
 s
 o
 
 
 
 2
 k
 +
 1
 
 
 ,
 
 D
 
 k
 
 
 =
 
 
 
 s
 o
 
 
 
 2
 k
 
 
 .
 
 
 {\displaystyle B_{k}={\mathfrak {so}}_{2k+1},D_{k}={\mathfrak {so}}_{2k}.}

Thus, O(2k+1) = SO(2k+1) × {±I},<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
while 
 
 
 
 O
 ⁡
 (
 2
 k
 )
 ≠
 SO
 ⁡
 (
 2
 k
 )
 ×
 {
 ±
 I
 }
 
 
 {\displaystyle \operatorname {O} (2k)\neq \operatorname {SO} (2k)\times \{\pm I\}}
 
 and is instead a non-trivial <a href="/facts/Group_extension/pGoe0l4d">central extension</a> of PO(2k).
Beware that PO(2k+1) is isometries of RP2k = P(R2k+1), while PO(2k) is isometries of RP2k−1 = P(R2k) – 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.

In odd dimension, 
 
 
 
 SO
 ⁡
 (
 2
 k
 +
 1
 )
 ≅
 PSO
 ⁡
 (
 2
 k
 +
 1
 )
 =
 PO
 ⁡
 (
 2
 k
 +
 1
 )
 ,
 
 
 {\displaystyle \operatorname {SO} (2k+1)\cong \operatorname {PSO} (2k+1)=\operatorname {PO} (2k+1),}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> so the group of projective isometries can be identified with the group of rotational isometries.
In even dimension, SO(2k) → PSO(2k) and O(2k) → PO(2k) are both 2-to-1 covers, and PSO(2k) < PO(2k) is an <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> 2 subgroup.

<h3>General properties</h3>
PSO and PO are <a href="/facts/Centerless/gqAg6B8I">centerless</a>, as with PSL and PGL; this is because scalar matrices are not only the center of SO and O, but also the <a href="/facts/Hypercenter/Hw9lbKGf">hypercenter</a> (quotient by the center does not always yield a centerless group).
PSO is the <a href="/facts/Maximal_compact_subgroup/8EA7XRcu">maximal compact subgroup</a> in the <a href="/facts/Projective_special_linear_group/y7P9rMyR">projective special linear group</a> PSL, while PO is maximal compact in the <a href="/facts/Projective_general_linear_group/y7P9rMyR">projective general linear group</a> PGL. This is analogous to SO being maximal compact in SL and O being maximal compact in GL.

<h2 id="representation-theory">Representation theory</h2>
Further information: <a href="/facts/Projective_representation/qva9eFMC">Projective representation</a>
PO is of basic interest in representation theory: a group homomorphism G → PGL is called a <a href="/facts/Projective_representation/qva9eFMC">projective representation</a> of G, just as a map G → GL is called a linear representation of G, and just as any linear representation can be reduced to a map G → O (by taking an invariant inner product), any projective representation can be reduced to a map G → PO.
See <a href="/facts/Projective_linear_group/y7P9rMyR">projective linear group: representation theory</a> for further discussion.

<h2 id="subgroups">Subgroups</h2>
See also: <a href="/facts/Spin_group/tfvR9cqM">Spin group § Discrete subgroups</a>, and <a href="/facts/Binary_polyhedral_group/Uc4MRIyl">Binary polyhedral group</a>
Subgroups of the projective orthogonal group correspond to subgroups of the orthogonal group that contain −I (that have <a href="/facts/Central_symmetry/UzswAm8N">central symmetry</a>). As always with a quotient map (by the <a href="/facts/Lattice_theorem/aJKEaMHC">lattice theorem</a>), there is a <a href="/facts/Galois_connection/lwuFXX2d">Galois connection</a> 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 <a href="/facts/Projective_polytope/dJGo1hGD">projective polytopes</a> – these correspond to the (discrete) point groups that include central symmetry. Compare with <a href="/facts/Spin_group/tfvR9cqM">discrete subgroups of the Spin group</a>, particularly the 3-dimensional case of <a href="/facts/Binary_polyhedral_group/Uc4MRIyl">binary polyhedral groups</a>.
For example, in 3 dimensions, 4 of the 5 <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a> have central symmetry (cube/octahedron, dodecahedron/icosahedron), while the tetrahedron does not – however, the <a href="/facts/Stellated_octahedron/oNkHNB3C">stellated octahedron</a> has central symmetry, though the resulting symmetry group is the same as that of the cube/octahedron.

<h2 id="topology">Topology</h2>
See also: <a href="/facts/Orthogonal_group/jKWM1KDM">Orthogonal group § Covering and quotient groups</a>
PO and PSO, as centerless topological groups, are at the bottom of a sequence of <a href="/facts/Covering_group/IrRjtTL2">covering groups</a>, whose top are the (<a href="/facts/Simply_connected/xFlta63J">simply connected</a>) <a href="/facts/Pin_group/1lHT68gC">Pin groups</a> or <a href="/facts/Spin_group/tfvR9cqM">Spin group</a>, respectively:

Pin±(n) → O(n) → PO(n).
Spin(n) → SO(n) → PSO(n).
These groups are all <a href="/facts/Compact_real_form/7Frha9gL">compact real forms</a> of the same Lie algebra.
These are all 2-to-1 covers, except for SO(2k+1) → PSO(2k+1) which is 1-to-1 (an isomorphism).

<h3>Homotopy groups</h3>
<a href="/facts/Homotopy_group/xjogL95Y">Homotopy groups</a> above 
 
 
 
 
 π
 
 1
 
 
 
 
 {\displaystyle \pi _{1}}
 
 do not change under covers, so they agree with those of the orthogonal group. The lower homotopy groups are given as follows.

π
          
            0
          
        
        (
        PSO
        )
        ≅
        1
      
    
    {\displaystyle \pi _{0}(\operatorname {PSO} )\cong 1}

π
          
            0
          
        
        (
        PO
        ⁡
        (
        2
        k
        )
        )
        ≅
        
          Z
        
        
          /
        
        2
        
          Z
        
        ,
        
          π
          
            0
          
        
        (
        PO
        ⁡
        (
        2
        k
        +
        1
        )
        )
        ≅
        1.
      
    
    {\displaystyle \pi _{0}(\operatorname {PO} (2k))\cong \mathbf {Z} /2\mathbf {Z} ,\pi _{0}(\operatorname {PO} (2k+1))\cong 1.}

The fundamental group of (centerless) PSO(n) equals the center of (simply connected) Spin(n), which is always true about covering groups:

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        n
        )
        )
        =
        
          π
          
            1
          
        
        (
        PO
        ⁡
        (
        n
        )
        )
        =
        Z
        ⁡
        (
        Spin
        ⁡
        (
        n
        )
        )
        .
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (n))=\pi _{1}(\operatorname {PO} (n))=\operatorname {Z} (\operatorname {Spin} (n)).}

Using the <a href="/facts/Spin_group/tfvR9cqM">table of centers of Spin groups</a> yields (for 
 
 
 
 k
 ≥
 1
 
 
 {\displaystyle k\geq 1}
 
):

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        4
        k
        )
        )
        =
        
          Z
        
        
          /
        
        2
        
          Z
        
        ⊕
        
          Z
        
        
          /
        
        2
        
          Z
        
        ,
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (4k))=\mathbf {Z} /2\mathbf {Z} \oplus \mathbf {Z} /2\mathbf {Z} ,}

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        4
        k
        +
        2
        )
        )
        =
        
          Z
        
        
          /
        
        4
        
          Z
        
        ,
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (4k+2))=\mathbf {Z} /4\mathbf {Z} ,}

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        2
        k
        +
        1
        )
        )
        =
        
          π
          
            1
          
        
        (
        SO
        ⁡
        (
        2
        k
        +
        1
        )
        )
        =
        
          Z
        
        
          /
        
        2
        
          Z
        
        ,
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (2k+1))=\pi _{1}(\operatorname {SO} (2k+1))=\mathbf {Z} /2\mathbf {Z} ,}

In low dimensions:

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        1
        )
        )
        =
        1
        ,
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (1))=1,}
  
 as the group is trivial.

π
          
            1
          
        
        (
        PSO
        ⁡
        (
        2
        )
        )
        =
        
          Z
        
        ,
      
    
    {\displaystyle \pi _{1}(\operatorname {PSO} (2))=\mathbf {Z} ,}
  
 as it is topologically a circle, though note that the preimage of the identity in Spin(2) is 
  
    
      
        
          Z
        
        
          /
        
        4
        
          Z
        
        ,
      
    
    {\displaystyle \mathbf {Z} /4\mathbf {Z} ,}
  
 as for other 
  
    
      
        4
        k
        +
        2.
      
    
    {\displaystyle 4k+2.}

<h3>Homology groups</h3>

<h3>Bundles</h3>

Just as the orthogonal group is the <a href="/facts/Structure_group/SJ7Ek6cI">structure group</a> of <a href="/facts/Vector_bundle/cXACAa1I">vector bundles</a>, the projective orthogonal group is the structure group of <a href="/facts/Projective_bundle/W7jjGmlW">projective bundles</a>, and the corresponding <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> is denoted BPO.

<h2 id="generalizations">Generalizations</h2>
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 <a href="/facts/Indefinite_form/utlBuVpN">indefinite forms</a>, and are denoted PO(p,q) by their signature.
The complex projective orthogonal group, PO(n,C) should not be confused with the <a href="/facts/Projective_unitary_group/LsQB1hH3">projective unitary group</a>, PU(n): PO preserves a symmetric form, while PU preserves a <a href="/facts/Hermitian_form/Kp6qUA8v">hermitian form</a> – PU is the symmetries of complex projective space (preserving the <a href="/facts/Fubini%25E2%2580%2593Study_metric/1uemCFdc">Fubini–Study metric</a>).
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 <a href="/facts/Orthogonal_group/jKWM1KDM">Dickson invariant</a>.

<h3>Finite fields</h3>

The projective orthogonal group over a finite field is used in the construction of a family of finite <a href="/facts/Simple_group/e5hGF7nT">simple groups</a> of <a href="/facts/Group_of_Lie_type/gmboug78">Lie type</a>, namely the <a href="/facts/Chevalley_group/gmboug78">Chevalley groups</a> of type Dn. 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 <a href="/facts/Spinor_norm/jKWM1KDM">spinor norm</a> (in odd characteristic) or the quasideterminant (in even characteristic).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The quasideterminant can be defined as (−1)D, where D is the <a href="/facts/Orthogonal_group/jKWM1KDM">Dickson invariant</a> (it is the determinant defined by the Dickson invariant), or in terms of the dimension of the fixed space.

<h2 id="notes">Notes</h2>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Projective_linear_group/y7P9rMyR">Projective linear group</a></li>
<li><a href="/facts/Projective_unitary_group/LsQB1hH3">Projective unitary group</a></li>
<li><a href="/facts/Orthogonal_group/jKWM1KDM">Orthogonal group</a></li>
<li><a href="/facts/Spin_group/tfvR9cqM">Spin group</a></li></ul>

<ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">Conway, John Horton</a>; Smith, Derek Alan (2003-02-07), "3.7 The Projective or Elliptic Groups", On quaternions and octonions, A K Peters, Ltd., pp. <a href="https://books.google.com/books?id=E_HCwwxMbfMC&pg=PA34">34</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-134-5</li>
<li>Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. "The Groups GOn(q), SOn(q), PGOn(q), and PSOn(q), and On(q)." §2.4 in Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, pp. xi–xii, 1985.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ProjectiveGeneralOrthogonalGroup.html">"Projective General Orthogonal Group"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/ProjectiveSpecialOrthogonalGroup.html">"Projective Special Orthogonal Group"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">A quadratic space is a vector space V together with a quadratic form Q; the Q is dropped from notation when it is clear. <a href="/wiki/Vector_space" target="_blank">/wiki/Vector_space</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">This product is an internal direct sum – a product of subgroups – not just an abstract external direct sum. <a href="/wiki/Internal_direct_sum" target="_blank">/wiki/Internal_direct_sum</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">The isomorphism/equality distinction in this equation is because the context is the 2-to-1 quotient map O → PO – PSO(2k+1) and PO(2k+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 (Conway & Smith 2003, p. 34) for an example of this distinction being made. <a href="/wiki/Isomorphism" target="_blank">/wiki/Isomorphism</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">ATLAS, p. xi <a href="/wiki/ATLAS_of_Finite_Groups" target="_blank">/wiki/ATLAS_of_Finite_Groups</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Projective orthogonal group open-in-new

Projective orthogonal group