Namespaces
Variants
Actions

Difference between revisions of "Orthogonal group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(Comments: link to Pythagorean field)
 
(4 intermediate revisions by one other user not shown)
Line 1: Line 1:
The group of all linear transformations of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703001.png" />-dimensional [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703002.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703003.png" /> which preserve a fixed non-singular [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703004.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703005.png" /> (i.e. linear transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703006.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703007.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703008.png" />). An orthogonal group is a [[Classical group|classical group]]. The elements of an orthogonal group are called orthogonal transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o0703009.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030010.png" />), or also automorphisms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030011.png" />. Furthermore, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030012.png" /> (for orthogonal groups over fields with characteristic 2 see [[#References|[1]]], [[#References|[7]]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030013.png" /> be the non-singular symmetric bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030014.png" /> related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030015.png" /> by the formula
+
{{MSC|20G}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030016.png" /></td> </tr></table>
 
  
The orthogonal group then consists of those linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030017.png" /> that preserve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030018.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030019.png" />, or (when one is talking of a specific field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030020.png" /> and a specific form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030021.png" />) simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030023.png" /> is the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030024.png" /> with respect to some basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030025.png" />, then the orthogonal group can be identified with the group of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030026.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030027.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030030.png" /> is transposition).
+
An orthogonal group is a group of all linear transformations of an $n$-dimensional
 +
[[Vector space|vector space]] $V$ over a field $k$ which preserve a fixed non-singular
 +
[[Quadratic form|quadratic form]] $Q$ on $V$ (i.e. linear transformations $\def\phi{\varphi}\phi$ such that $Q(\phi(v))=Q(v)$ for all $v\in V$). An orthogonal group is a
 +
[[Classical group|classical group]]. The elements of an orthogonal
 +
group are called orthogonal transformations of $V$ (with respect to
 +
$Q$), or also automorphisms of the form $Q$. Furthermore, let
 +
${\rm char\;} k\ne 2$ (for orthogonal groups over fields with characteristic 2 see
 +
{{Cite|Di}},
 +
{{Cite|2}}) and let $f$ be the non-singular symmetric bilinear form on $V$ related to $Q$ by the formula
  
The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030031.png" /> is equal to 1 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030032.png" />. Elements with determinant 1 are called rotations; they form a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030033.png" /> (or simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030034.png" />) of index 2 in the orthogonal group, called the rotation group. Elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030035.png" /> are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030036.png" />.
+
$$f(u,v)=\frac{1}{2}(Q(u+v) - Q(u) - Q(v)).$$
 +
The orthogonal group then consists of those linear transformations of
 +
$V$ that preserve $f$, and is denoted by $\def\O{ {\rm O} }\O_n(k,f)$, or (when one is talking of a specific field $k$ and a specific form $f$) simply by $\O_n$. If $B$ is the matrix of $f$ with respect to some basis of $V$, then the orthogonal group can be identified with the group of all $(n\times n)$-matrices $A$ with coefficients in $k$ such that $A^TBA = B$ (${}^T$ is transposition).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030037.png" /> be the group of all homotheties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030040.png" />, of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030041.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030042.png" /> is the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030043.png" />; it consists of two elements: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030046.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030047.png" /> is the direct product of its centre and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030048.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030049.png" />, the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030050.png" /> is trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030051.png" /> is odd, and coincides with the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030052.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030053.png" /> is even. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030054.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030055.png" /> is commutative and is isomorphic either to the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030056.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030057.png" /> (when the Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030059.png" /> is equal to 1), or to the group of elements with norm 1 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030060.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030061.png" /> is the discriminant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030062.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030063.png" />). The commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030064.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030065.png" />, or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030066.png" />; it is generated by the squares of the elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030067.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030068.png" />, the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030069.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030070.png" />. The centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030071.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030072.png" />.
+
The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from $\O_n$ is equal to 1 or $-1$. Elements with determinant 1 are called rotations; they form a normal subgroup $\O_n^+(k,f)$ (or simply $\O_n^+$) of index 2 in the orthogonal group, called the rotation group. Elements from $\O_n\setminus \O_n^+$ are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from $\O_n$.
  
Other classical groups related to orthogonal groups include the canonical images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030074.png" /> in the [[Projective group|projective group]]; they are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030076.png" /> (or simply by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030078.png" />) and are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030080.png" />, respectively.
+
Let $Z_n$ be the group of all homotheties $\def\a{\alpha}\phi_\a : v\mapsto \a v$, $\a\in k$, $\a\ne 0$, of the space $V$. Then $\O_n\cap Z_n$ is the centre of $\O_n$; it consists of two elements: $\phi_1$ and $\phi_{-1}$. If $n$ is odd, then $\O_n$ is the direct product of its centre and $\O_n^+$. If $n\ge 3$, the centre of $\O_n^+$ is trivial if $n$ is odd, and coincides with the centre of $\O_n$ if $n$ is even. If $n=2$, the group $\O_n^+$ is commutative and is isomorphic either to the multiplicative group $k^*$ of $k$ (when the Witt index $\nu$ of $f$ is equal to 1), or to the group of elements with norm 1 in $k(\sqrt-\Delta)$, where $\Delta$ is the discriminant of $f$ (when $\nu=0$). The commutator subgroup of $\O_n(k,f)$ is denoted by $\def\Om{\Omega}\Om_n(k,f)$, or simply by $\Om_n$; it is generated by the squares of the elements from $\O_n$. When $n\ge 3$, the commutator subgroup of $\O_n^+$ coincides with $\Om_n$. The centre of $\Om_n$ is $\Om_n\cap Z_n$.
 +
 
 +
Other classical groups related to orthogonal groups include the canonical images of $\O_n^+$ and $\Om_n$ in the
 +
[[Projective group|projective group]]; they are denoted by ${\rm P}\O_n^+(k,f)$ and ${\rm P}\Om_n(k,f)$ (or simply by ${\rm P}\O_n^+$ and ${\rm P}\Om_n$) and are isomorphic to $\O_n^+/(\O_n^+\cap Z_n)$ and $\Om_n/(\Om_n\cap Z_n)$, respectively.
  
 
The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:
 
The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030081.png" /></td> </tr></table>
+
$$\O_n\supset \O_n^+\supset \Om_n\supset \Om_n\cap Z_n \supset \{e\}.$$
 
+
The group $\O_n/\O_n^+$ has order 2. Every element in $\O_n/\Om_n$ has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form $2^\a$ where $\a$ is an integer. The description of the remaining factors depends essentially on the Witt index $\nu$ of the form $f$.
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030082.png" /> has order 2. Every element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030083.png" /> has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030084.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030085.png" /> is an integer. The description of the remaining factors depends essentially on the Witt index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030086.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030087.png" />.
 
 
 
First, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030088.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030089.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030090.png" />. This isomorphism is defined by the spinor norm, which defines an epimorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030091.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030092.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030093.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030094.png" /> is non-trivial (and consists of the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030096.png" />) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030097.png" /> is even and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030098.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o07030099.png" />, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300100.png" /> is simple. The cases where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300101.png" /> are studied separately. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300102.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300103.png" /> (see [[Special linear group|Special linear group]]) and is also simple if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300104.png" /> has at least 4 elements (the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300105.png" /> is isomorphic to the projective group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300106.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300107.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300108.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300109.png" /> and is simple (in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300110.png" />), while when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300111.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300112.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300113.png" /> and is not simple. In the particular case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300114.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300115.png" /> is a form of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300116.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300117.png" /> is called the Lorentz group.
 
 
 
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300118.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300119.png" /> is an anisotropic form), these results are not generally true. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300121.png" /> is a positive-definite form, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300122.png" />, although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300123.png" /> consists of two elements; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300125.png" />, one can have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300126.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300127.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300128.png" />, the structures of an orthogonal group and its related groups essentially depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300129.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300130.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300134.png" />, is simple (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300135.png" /> is isomorphic to the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300136.png" /> of two simple groups); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300137.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300138.png" />-adic numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300139.png" />, there exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300140.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300141.png" />) an infinite normal series with Abelian quotients. Important special cases are when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300142.png" /> is a locally compact field or an algebraic number field. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300143.png" /> is the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300144.png" />-adic numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300145.png" /> is impossible when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300146.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300147.png" /> is an algebraic number field, then there is no such restriction and one of the basic results is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300148.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300150.png" />, is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300151.png" /> by extension of coefficients to the local fields defined by valuations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300152.png" /> (the [[Hasse principle|Hasse principle]]).
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300153.png" /> is the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300154.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300155.png" /> elements, then an orthogonal group is finite. The order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300156.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300157.png" /> odd is equal to
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300158.png" /></td> </tr></table>
 
 
 
while when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300159.png" /> it is equal to
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300160.png" /></td> </tr></table>
+
First, let $\nu\ge 1$. Then $\O_n^+/\Om_n \simeq k^*/{k^*}^2$ when $n>2$. This isomorphism is defined by the spinor norm, which defines an epimorphism from $\O_n^+$ on $k^*/{k^*}^2$ with kernel $\Om_n$. The group $\Om_n\cap Z_n$ is non-trivial (and consists of the transformations $\phi_1$ and $\phi_{-1}$) if and only if $n$ is even and $\Delta\in {k^*}^2$. If $n\ge 5$, then the group ${\rm P}\Om_n = \Om_n/(\Om_n\cap Z_n)$ is simple. The cases where $n=3,4$ are studied separately. Namely, ${\rm P}\Om_3 = \Om_3$ is isomorphic to $\def\PSL{ {\rm PSL}}\PSL_2(k)$ (see
 +
[[Special linear group|Special linear group]]) and is also simple if $k$ has at least 4 elements (the group $\O_3^+$ is isomorphic to the projective group $\def\PGL{ {\rm PGL}}\PGL_2(k)$). When $\nu=1$, the group ${\rm P}\Om_4 = \Om_4$ is isomorphic to the group $\PSL_2(k(\sqrt{\Delta}))$ and is simple (in this case $\Delta\notin k^2$), while when $\nu=2$, the group ${\rm P}\Om_4$ is isomorphic to $\PSL_2(k)\times \PSL_2(k)$ and is not simple. In the particular case when $k = \R$ and $Q$ is a form of signature $(3,1)$, the group ${\rm P}\Om_4 = \Om_4\simeq \PSL_2(\C)$ is called the [[Lorentz group]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300161.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300163.png" /> otherwise. These formulas and general facts about orthogonal groups when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300164.png" /> also allow one to calculate the orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300166.png" />, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300167.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300168.png" />, while the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300169.png" /> is equal to 2. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300170.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300171.png" />, is one of the classical simple finite groups (see also [[Chevalley group|Chevalley group]]).
+
When $\nu = 0$ (i.e. $Q$ is an anisotropic form), these results are
 +
not generally true. For example, if $k=\R$ and $Q$ is a
 +
positive-definite form, then $\Om_n = O_n^+$, although $\R^*/{\R^*}^2$
 +
consists of two elements; when $k=\Q$, $n=4$, one can have $\Delta\in
 +
k^2$, but $\phi_{-1}\notin \Om_4$. When $\nu=0$, the structures of an
 +
orthogonal group and its related groups essentially depend on $k$. For
 +
example, if $k=\R$, then ${\rm P}\O_n^+$, $n\ge 3$, $n\ne 4$, $\nu=0$, is simple (and ${\rm P}\O_4^+$ is isomorphic to the direct product $\O_3^+ \times \O_3^+$ of two simple groups); if $k$ is the field of $p$-adic numbers and $\nu=0$, there exists in $\O_3$ (and $\O_4$) an infinite normal series with Abelian quotients. Important special cases are when $k$ is a locally compact field or an algebraic number field. If $k$ is the field of $p$-adic numbers, then $n=0$ is impossible when $\nu\ge 5$. If $k$ is an algebraic number field, then there is no such restriction and one of the basic results is that ${\rm P}\Om_n$, when $\nu=0$ and $n\ge 5$, is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from $Q$ by extension of coefficients to the local fields defined by valuations of $k$ (the
 +
[[Hasse principle|Hasse principle]]).
  
One of the basic results on automorphisms of orthogonal groups is the following: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300172.png" />, then every automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300173.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300174.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300176.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300177.png" /> is a fixed homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300178.png" /> into its centre and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300179.png" /> is a fixed bijective semi-linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300180.png" /> onto itself satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300181.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300182.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300183.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300184.png" /> is an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300185.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300186.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300187.png" />, then every automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300188.png" /> is induced by an automorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300189.png" /> (see [[#References|[1]]], [[#References|[3]]]).
+
If $k$ is the finite field $\F_q$ of $q$ elements, then an orthogonal group is finite. The order of $\O_n^+$ for $n$ odd is equal to
  
Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300190.png" /> be an anisotropic form such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300191.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300192.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300193.png" /> is a Pythagorean orderable field. For a fixed order of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300194.png" />, any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300195.png" /> constructed from a linearly independent basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300196.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300197.png" /> is the set of all linear combinations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300198.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300199.png" />, is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300200.png" />-dimensional chain of incident half-spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300201.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300202.png" /> has the property of free mobility, i.e. for any two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300203.png" />-dimensional chains of half-spaces there exists a unique transformation from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300204.png" /> which transforms the first chain into the second. This property characterizes an orthogonal group: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300205.png" /> is any ordered skew-field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300206.png" /> is a subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300207.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300208.png" />, having the property of free mobility, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300209.png" /> is a Pythagorean field, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300210.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300211.png" /> is an anisotropic symmetric bilinear form such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300212.png" /> for any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300213.png" />.
+
$$(q^{n-1}-1)q^{n-2}(q^{n-3}-1)q^{n-4}\cdots (q^2-1)q,$$
 +
while when $n=2m$ it is equal to
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300214.png" /> be a fixed algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300215.png" />. The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300216.png" /> extends naturally to a non-singular symmetric bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300217.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300218.png" />, and the orthogonal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300219.png" /> is a [[Linear algebraic group|linear algebraic group]] defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300220.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300221.png" /> as group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300222.png" />-points. The linear algebraic groups thus defined (for various <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300223.png" />) are isomorphic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300224.png" /> (but in general not over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300225.png" />); the corresponding linear algebraic group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300226.png" /> is called the orthogonal algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300227.png" />. Its subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300228.png" /> is also a linear algebraic group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300229.png" />, and is called a properly orthogonal, or special orthogonal algebraic group (notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300230.png" />); it is the connected component of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300231.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300232.png" /> is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300233.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300234.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300235.png" />, and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300236.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300237.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300238.png" />. The universal covering group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300239.png" /> is a [[Spinor group|spinor group]].
+
$$\def\e{\epsilon}(q^{2m-1}-\e q^{m-1})(q^{2m-2}-1)q^{2m-3}\cdots(q^2-1)q,$$
 +
where $\e=1$ if $(-1)^m\Delta\in \F_q^2$ and $\e=-1$ otherwise. These formulas and general facts about orthogonal groups when $\nu\ge 1$ also allow one to calculate the orders of $\Om_n$ and ${\rm P}\Om_n$, since $\nu\ge 1$ when $n\ge 3$, while the order of $k^*/{k^*}^2$ is equal to 2. The group ${\rm P}\Om_n$, $n\ge 5$, is one of the classical simple finite groups (see also
 +
[[Chevalley group|Chevalley group]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300240.png" /> or a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300241.png" />-adic field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300242.png" /> has a canonical structure of a real, complex or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300243.png" />-adic [[Analytic group|analytic group]]. The Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300244.png" /> is defined up to isomorphism by the signature of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300245.png" />; if this signature is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300246.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300247.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300248.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300249.png" /> and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300250.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300251.png" /> which satisfy
+
One of the basic results on automorphisms of orthogonal groups is the
 +
following: If $n\ge 3$, then every automorphism $\phi$ of $\O_n$ has
 +
the form $\phi(u)=\chi(u)gug^{-1}$, $u\in \O_n$, where $\chi$ is a
 +
fixed homomorphism of $\O_n$ into its centre and $g$ is a fixed
 +
bijective semi-linear mapping of $V$ onto itself satisfying
 +
$Q(g(v))=r_gQ^\sigma(v)$ for all $v\in V$, where $r_g\in k^*$ while
 +
$\sigma$ is an automorphism of $k$. If $\nu\ge 1$ and $n\ge 6$, then
 +
every automorphism of $\O_n^+$ is induced by an automorphism of $\O_n$ (see
 +
{{Cite|Di}},
 +
{{Cite|}}).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300252.png" /></td> </tr></table>
+
Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let $ Q$ be an anisotropic form such that $Q(v)\in k^2$ for all $v\in V$. In this case $k$ is a Pythagorean orderable field. For a fixed order of the field $k$, any sequence $((H_s)_{1\le s\le n}$ constructed from a linearly independent basis $((h_s)_{1\le s\le n}$, where $H_s$ is the set of all linear combinations of the form $\def\l{\lambda}\sum_{j=1}^sl_jh_j$, $\l_s\ge 0$, is called an $n$-dimensional chain of incident half-spaces in $V$. The group $\O_n$ has the property of free mobility, i.e. for any two $n$-dimensional chains of half-spaces there exists a unique transformation from $\O_n$ which transforms the first chain into the second. This property characterizes an orthogonal group: If $L$ is any ordered skew-field and $G$ is a subgroup in ${\rm GL_n(L)}$, $n\ge 3$, having the property of free mobility, then $L$ is a Pythagorean field, while $G=\O_n(L,f)$, where $f$ is an anisotropic symmetric bilinear form such that $f(v,v)\in L_1^2$ for any vector $v$.
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300253.png" /> denotes the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300254.png" />-matrix). The Lie algebra of this group is the Lie algebra of all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300255.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300256.png" /> that satisfy the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300257.png" />. In the particular case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300258.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300259.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300260.png" /> and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300261.png" />-matrices. The Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300262.png" /> has four connected components when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300263.png" />, and two connected components when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300264.png" />. The connected component of the identity is its commutator subgroup, which, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300265.png" />, coincides with the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300266.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300267.png" /> consisting of all transformations with determinant 1. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300268.png" /> is compact only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300269.png" />. The topological invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300270.png" /> have been studied. One of the classical results is the calculation of the Betti numbers of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300271.png" />: Its Poincaré polynomial has the form
+
Let $\def\bk{ {\bar k}}\bk$ be a fixed algebraic closure of the field $k$. The form $f$ extends naturally to a non-singular symmetric bilinear form ${\bar f}$ on $V\otimes_k \bk$, and the orthogonal group $\O_n(\bk,f)$ is a
 +
[[Linear algebraic group|linear algebraic group]] defined over $k$ with $\O_n(k,f)$ as group of $k$-points. The linear algebraic groups thus defined (for various $f$) are isomorphic over $\bk$ (but in general not over $k$); the corresponding linear algebraic group over $\bk$ is called the orthogonal algebraic group $\O_n(\bk)$. Its subgroup $\O_n^+(\bk,{\bar f})$ is also a linear algebraic group over $\bk$, and is called a properly orthogonal, or special orthogonal algebraic group (notation: $\def\SO{ {\rm SO}}\SO_n(\bk)$); it is the connected component of the identity of $\O_n(\bk)$. The group $\SO_n(\bk)$ is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type $B_s$ when $n=2s+1$, $s\ge 1$, and of type $D_s$ when $n=2s$, $s\ge 3$. The universal covering group of $\SO_n$ is a
 +
[[Spinor group|spinor group]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300272.png" /></td> </tr></table>
+
If $ k=\R,\C$ or a $p$-adic field, then $\O_n(k,f)$ has a canonical structure of a real, complex or $p$-adic
 +
[[Analytic group|analytic group]]. The Lie group $\O_n(\R,f)$ is defined up to isomorphism by the signature of the form $f$; if this signature is $(p,q)$, $p+q=n$, then $\O_n(\R,f)$ is denoted by $\O_(p,q)$ and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real $(n\times n)$-matrices $A$ which satisfy
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300273.png" />, and the form
+
$$A^TI_{p,q}A = I_{p,q}\qquad\textrm{ where }I_{p,q} = \begin{pmatrix}1_p & 0 \\ 0 & -1_q\end{pmatrix}$$
 +
($1_s$ denotes the unit $(s\times s)$-matrix). The Lie algebra of this group is the Lie algebra of all real $(n\times n)$-matrices $X$ that satisfy the condition $X^TI_{p,q} = -I_{p.q}X$. In the particular case $q=0$, the group $\O(p,q)$ is denoted by $\O(n)$ and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real $(n\times n)$-matrices. The Lie group $\O(p,q)$ has four connected components when $q\ne 0$, and two connected components when $q=0$. The connected component of the identity is its commutator subgroup, which, when $q=0$, coincides with the subgroup $\def\SO{ {\rm SO}}\SO(n)$ in $\O(n)$ consisting of all transformations with determinant 1. The group $\O(p,q)$ is compact only when $q=0$. The topological invariants of $\SO(n)$ have been studied. One of the classical results is the calculation of the Betti numbers of the manifold $\SO(n)$: Its Poincaré polynomial has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300274.png" /></td> </tr></table>
+
$$\prod_{s=1}^m(1+t^{4s-1})$$
 +
when $n=2m+1$, and the form
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300275.png" />. The fundamental group of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300276.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300277.png" />. The calculation of the higher homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300278.png" /> is directly related to the classification of locally trivial principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300279.png" />-fibrations over spheres. An important part in topological [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300280.png" />-theory]] is played by the periodicity theorem, according to which, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300281.png" />, there are the isomorphisms
+
$$(1+t^{2m-1})\prod_{s=1}^{m-1}(1+t^{4s-1})$$
 
+
when $n=2m$. The fundamental group of the manifold $\SO(n)$ is $\Z_2$. The calculation of the higher homotopy groups $\pi_l(\SO(n))$ is directly related to the classification of locally trivial principal $\SO(n)$-fibrations over spheres. An important part in topological
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300282.png" /></td> </tr></table>
+
[[K-theory|$K$-theory]] is played by the periodicity theorem, according to which, when $N\gg n$, there are the isomorphisms
  
 +
$$\pi_{n+8}(\O(N)) \simeq \pi_{n}(\O(N));$$
 
further,
 
further,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300283.png" /></td> </tr></table>
+
$$\pi_n(\O(N)) \simeq \Z_2$$
 
+
if $n=0,1$;
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300284.png" />;
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300285.png" /></td> </tr></table>
 
 
 
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300286.png" />; and
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300287.png" /></td> </tr></table>
 
 
 
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300288.png" />. The study of the topology of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300289.png" /> reduces in essence to the previous case, since the connected component of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300290.png" /> is diffeomorphic to the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070300/o070300291.png" /> on a Euclidean space.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.A. Dieudonné,  "La géométrie des groups classiques" , Springer  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Artin,  "Geometric algebra" , Interscience  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''Automorphisms of the classical groups'' , Moscow  (1976)  (In Russian; translated from English and French)  (Collection of translations)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Weyl,  "The classical groups, their invariants and representations" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from French)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  O.T. O'Meara,  "Introduction to quadratic forms" , Springer  (1973)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>
 
  
 +
$$\pi_n(\O(N)) \simeq \Z$$
 +
if $n=3,7$; and
  
 +
$$\pi_n(\O(N)) = 0$$
 +
if $n=2,4,5,6$. The study of the topology of the group $\O(p,q)$ reduces in essence to the previous case, since the connected component of the identity of $\O(p,q)$ is diffeomorphic to the product $\SO(p)\times \SO(q)$ on a Euclidean space.
  
 
====Comments====
 
====Comments====
A Pythagorean field is a field in which the sum of two squares is again a square.
+
A [[Pythagorean field]] is a field in which the sum of two squares is again a square.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Dieudonné,   "On the automorphisms of the classical groups" , ''Mem. Amer. Math. Soc.'' , '''2''' , Amer. Math. Soc. (1951)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "Geometric algebra", Interscience (1957) {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.02101}}
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", '''2''', Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2272929}} {{MR|0928386}} {{MR|0896478}} {{MR|0782297}} {{MR|0782296}} {{MR|0722608}} {{MR|0682756}} {{MR|0643362}} {{MR|0647314}} {{MR|0610795}} {{MR|0583191}} {{MR|0354207}} {{MR|0360549}} {{MR|0237342}} {{MR|0205211}} {{MR|0205210}} {{ZBL|}}
 +
|-
 +
  |valign="top"|{{Ref|Di}}||valign="top"| J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) {{MR|}} {{ZBL|0221.20056}}
 +
|-
 +
|valign="top"|{{Ref|Di2}}||valign="top"| J. Dieudonné, "On the automorphisms of the classical groups", ''Mem. Amer. Math. Soc.'', '''2''', Amer. Math. Soc. (1951) {{MR|0045125}} {{ZBL|0042.25603}}
 +
|-
 +
|valign="top"|{{Ref|Hu}}||valign="top"| D. Husemoller, "Fibre bundles", McGraw-Hill (1966) {{MR|0229247}} {{ZBL|0144.44804}}
 +
|-
 +
|valign="top"|{{Ref|OM}}||valign="top"| O.T. O'Meara, "Introduction to quadratic forms", Springer (1973) {{MR|}} {{ZBL|0259.10018}}
 +
|-
 +
|valign="top"|{{Ref|We}}||valign="top"| H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) {{MR|0000255}} {{ZBL|1024.20502}}
 +
|-
 +
|valign="top"|{{Ref|Zh}}||valign="top"| D.P. Zhelobenko, "Compact Lie groups and their representations", Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}}
 +
|-
 +
|}

Latest revision as of 19:20, 7 November 2014

2010 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]


An orthogonal group is a group of all linear transformations of an $n$-dimensional vector space $V$ over a field $k$ which preserve a fixed non-singular quadratic form $Q$ on $V$ (i.e. linear transformations $\def\phi{\varphi}\phi$ such that $Q(\phi(v))=Q(v)$ for all $v\in V$). An orthogonal group is a classical group. The elements of an orthogonal group are called orthogonal transformations of $V$ (with respect to $Q$), or also automorphisms of the form $Q$. Furthermore, let ${\rm char\;} k\ne 2$ (for orthogonal groups over fields with characteristic 2 see [Di], [2]) and let $f$ be the non-singular symmetric bilinear form on $V$ related to $Q$ by the formula

$$f(u,v)=\frac{1}{2}(Q(u+v) - Q(u) - Q(v)).$$ The orthogonal group then consists of those linear transformations of $V$ that preserve $f$, and is denoted by $\def\O{ {\rm O} }\O_n(k,f)$, or (when one is talking of a specific field $k$ and a specific form $f$) simply by $\O_n$. If $B$ is the matrix of $f$ with respect to some basis of $V$, then the orthogonal group can be identified with the group of all $(n\times n)$-matrices $A$ with coefficients in $k$ such that $A^TBA = B$ (${}^T$ is transposition).

The description of the algebraic structure of an orthogonal group is a classical problem. The determinant of any element from $\O_n$ is equal to 1 or $-1$. Elements with determinant 1 are called rotations; they form a normal subgroup $\O_n^+(k,f)$ (or simply $\O_n^+$) of index 2 in the orthogonal group, called the rotation group. Elements from $\O_n\setminus \O_n^+$ are called inversions. Every rotation (inversion) is the product of an even (odd) number of reflections from $\O_n$.

Let $Z_n$ be the group of all homotheties $\def\a{\alpha}\phi_\a : v\mapsto \a v$, $\a\in k$, $\a\ne 0$, of the space $V$. Then $\O_n\cap Z_n$ is the centre of $\O_n$; it consists of two elements: $\phi_1$ and $\phi_{-1}$. If $n$ is odd, then $\O_n$ is the direct product of its centre and $\O_n^+$. If $n\ge 3$, the centre of $\O_n^+$ is trivial if $n$ is odd, and coincides with the centre of $\O_n$ if $n$ is even. If $n=2$, the group $\O_n^+$ is commutative and is isomorphic either to the multiplicative group $k^*$ of $k$ (when the Witt index $\nu$ of $f$ is equal to 1), or to the group of elements with norm 1 in $k(\sqrt-\Delta)$, where $\Delta$ is the discriminant of $f$ (when $\nu=0$). The commutator subgroup of $\O_n(k,f)$ is denoted by $\def\Om{\Omega}\Om_n(k,f)$, or simply by $\Om_n$; it is generated by the squares of the elements from $\O_n$. When $n\ge 3$, the commutator subgroup of $\O_n^+$ coincides with $\Om_n$. The centre of $\Om_n$ is $\Om_n\cap Z_n$.

Other classical groups related to orthogonal groups include the canonical images of $\O_n^+$ and $\Om_n$ in the projective group; they are denoted by ${\rm P}\O_n^+(k,f)$ and ${\rm P}\Om_n(k,f)$ (or simply by ${\rm P}\O_n^+$ and ${\rm P}\Om_n$) and are isomorphic to $\O_n^+/(\O_n^+\cap Z_n)$ and $\Om_n/(\Om_n\cap Z_n)$, respectively.

The basic classical facts about the algebraic structure describe the successive factors of the following series of normal subgroups of an orthogonal group:

$$\O_n\supset \O_n^+\supset \Om_n\supset \Om_n\cap Z_n \supset \{e\}.$$ The group $\O_n/\O_n^+$ has order 2. Every element in $\O_n/\Om_n$ has order 2, thus this group is defined completely by its cardinal number, and this number can be either infinite or finite of the form $2^\a$ where $\a$ is an integer. The description of the remaining factors depends essentially on the Witt index $\nu$ of the form $f$.

First, let $\nu\ge 1$. Then $\O_n^+/\Om_n \simeq k^*/{k^*}^2$ when $n>2$. This isomorphism is defined by the spinor norm, which defines an epimorphism from $\O_n^+$ on $k^*/{k^*}^2$ with kernel $\Om_n$. The group $\Om_n\cap Z_n$ is non-trivial (and consists of the transformations $\phi_1$ and $\phi_{-1}$) if and only if $n$ is even and $\Delta\in {k^*}^2$. If $n\ge 5$, then the group ${\rm P}\Om_n = \Om_n/(\Om_n\cap Z_n)$ is simple. The cases where $n=3,4$ are studied separately. Namely, ${\rm P}\Om_3 = \Om_3$ is isomorphic to $\def\PSL{ {\rm PSL}}\PSL_2(k)$ (see Special linear group) and is also simple if $k$ has at least 4 elements (the group $\O_3^+$ is isomorphic to the projective group $\def\PGL{ {\rm PGL}}\PGL_2(k)$). When $\nu=1$, the group ${\rm P}\Om_4 = \Om_4$ is isomorphic to the group $\PSL_2(k(\sqrt{\Delta}))$ and is simple (in this case $\Delta\notin k^2$), while when $\nu=2$, the group ${\rm P}\Om_4$ is isomorphic to $\PSL_2(k)\times \PSL_2(k)$ and is not simple. In the particular case when $k = \R$ and $Q$ is a form of signature $(3,1)$, the group ${\rm P}\Om_4 = \Om_4\simeq \PSL_2(\C)$ is called the Lorentz group.

When $\nu = 0$ (i.e. $Q$ is an anisotropic form), these results are not generally true. For example, if $k=\R$ and $Q$ is a positive-definite form, then $\Om_n = O_n^+$, although $\R^*/{\R^*}^2$ consists of two elements; when $k=\Q$, $n=4$, one can have $\Delta\in k^2$, but $\phi_{-1}\notin \Om_4$. When $\nu=0$, the structures of an orthogonal group and its related groups essentially depend on $k$. For example, if $k=\R$, then ${\rm P}\O_n^+$, $n\ge 3$, $n\ne 4$, $\nu=0$, is simple (and ${\rm P}\O_4^+$ is isomorphic to the direct product $\O_3^+ \times \O_3^+$ of two simple groups); if $k$ is the field of $p$-adic numbers and $\nu=0$, there exists in $\O_3$ (and $\O_4$) an infinite normal series with Abelian quotients. Important special cases are when $k$ is a locally compact field or an algebraic number field. If $k$ is the field of $p$-adic numbers, then $n=0$ is impossible when $\nu\ge 5$. If $k$ is an algebraic number field, then there is no such restriction and one of the basic results is that ${\rm P}\Om_n$, when $\nu=0$ and $n\ge 5$, is simple. In this case, the study of orthogonal groups is closely connected with the theory of equivalence of quadratic forms, where one needs the forms obtained from $Q$ by extension of coefficients to the local fields defined by valuations of $k$ (the Hasse principle).

If $k$ is the finite field $\F_q$ of $q$ elements, then an orthogonal group is finite. The order of $\O_n^+$ for $n$ odd is equal to

$$(q^{n-1}-1)q^{n-2}(q^{n-3}-1)q^{n-4}\cdots (q^2-1)q,$$ while when $n=2m$ it is equal to

$$\def\e{\epsilon}(q^{2m-1}-\e q^{m-1})(q^{2m-2}-1)q^{2m-3}\cdots(q^2-1)q,$$ where $\e=1$ if $(-1)^m\Delta\in \F_q^2$ and $\e=-1$ otherwise. These formulas and general facts about orthogonal groups when $\nu\ge 1$ also allow one to calculate the orders of $\Om_n$ and ${\rm P}\Om_n$, since $\nu\ge 1$ when $n\ge 3$, while the order of $k^*/{k^*}^2$ is equal to 2. The group ${\rm P}\Om_n$, $n\ge 5$, is one of the classical simple finite groups (see also Chevalley group).

One of the basic results on automorphisms of orthogonal groups is the following: If $n\ge 3$, then every automorphism $\phi$ of $\O_n$ has the form $\phi(u)=\chi(u)gug^{-1}$, $u\in \O_n$, where $\chi$ is a fixed homomorphism of $\O_n$ into its centre and $g$ is a fixed bijective semi-linear mapping of $V$ onto itself satisfying $Q(g(v))=r_gQ^\sigma(v)$ for all $v\in V$, where $r_g\in k^*$ while $\sigma$ is an automorphism of $k$. If $\nu\ge 1$ and $n\ge 6$, then every automorphism of $\O_n^+$ is induced by an automorphism of $\O_n$ (see [Di], ).

Like the other classical groups, an orthogonal group has a geometric characterization (under certain hypotheses). Indeed, let $ Q$ be an anisotropic form such that $Q(v)\in k^2$ for all $v\in V$. In this case $k$ is a Pythagorean orderable field. For a fixed order of the field $k$, any sequence $((H_s)_{1\le s\le n}$ constructed from a linearly independent basis $((h_s)_{1\le s\le n}$, where $H_s$ is the set of all linear combinations of the form $\def\l{\lambda}\sum_{j=1}^sl_jh_j$, $\l_s\ge 0$, is called an $n$-dimensional chain of incident half-spaces in $V$. The group $\O_n$ has the property of free mobility, i.e. for any two $n$-dimensional chains of half-spaces there exists a unique transformation from $\O_n$ which transforms the first chain into the second. This property characterizes an orthogonal group: If $L$ is any ordered skew-field and $G$ is a subgroup in ${\rm GL_n(L)}$, $n\ge 3$, having the property of free mobility, then $L$ is a Pythagorean field, while $G=\O_n(L,f)$, where $f$ is an anisotropic symmetric bilinear form such that $f(v,v)\in L_1^2$ for any vector $v$.

Let $\def\bk{ {\bar k}}\bk$ be a fixed algebraic closure of the field $k$. The form $f$ extends naturally to a non-singular symmetric bilinear form ${\bar f}$ on $V\otimes_k \bk$, and the orthogonal group $\O_n(\bk,f)$ is a linear algebraic group defined over $k$ with $\O_n(k,f)$ as group of $k$-points. The linear algebraic groups thus defined (for various $f$) are isomorphic over $\bk$ (but in general not over $k$); the corresponding linear algebraic group over $\bk$ is called the orthogonal algebraic group $\O_n(\bk)$. Its subgroup $\O_n^+(\bk,{\bar f})$ is also a linear algebraic group over $\bk$, and is called a properly orthogonal, or special orthogonal algebraic group (notation: $\def\SO{ {\rm SO}}\SO_n(\bk)$); it is the connected component of the identity of $\O_n(\bk)$. The group $\SO_n(\bk)$ is an almost-simple algebraic group (i.e. does not contain infinite algebraic normal subgroups) of type $B_s$ when $n=2s+1$, $s\ge 1$, and of type $D_s$ when $n=2s$, $s\ge 3$. The universal covering group of $\SO_n$ is a spinor group.

If $ k=\R,\C$ or a $p$-adic field, then $\O_n(k,f)$ has a canonical structure of a real, complex or $p$-adic analytic group. The Lie group $\O_n(\R,f)$ is defined up to isomorphism by the signature of the form $f$; if this signature is $(p,q)$, $p+q=n$, then $\O_n(\R,f)$ is denoted by $\O_(p,q)$ and is called a pseudo-orthogonal group. It can be identified with the Lie group of all real $(n\times n)$-matrices $A$ which satisfy

$$A^TI_{p,q}A = I_{p,q}\qquad\textrm{ where }I_{p,q} = \begin{pmatrix}1_p & 0 \\ 0 & -1_q\end{pmatrix}$$ ($1_s$ denotes the unit $(s\times s)$-matrix). The Lie algebra of this group is the Lie algebra of all real $(n\times n)$-matrices $X$ that satisfy the condition $X^TI_{p,q} = -I_{p.q}X$. In the particular case $q=0$, the group $\O(p,q)$ is denoted by $\O(n)$ and is called a real orthogonal group; its Lie algebra consists of all skew-symmetric real $(n\times n)$-matrices. The Lie group $\O(p,q)$ has four connected components when $q\ne 0$, and two connected components when $q=0$. The connected component of the identity is its commutator subgroup, which, when $q=0$, coincides with the subgroup $\def\SO{ {\rm SO}}\SO(n)$ in $\O(n)$ consisting of all transformations with determinant 1. The group $\O(p,q)$ is compact only when $q=0$. The topological invariants of $\SO(n)$ have been studied. One of the classical results is the calculation of the Betti numbers of the manifold $\SO(n)$: Its Poincaré polynomial has the form

$$\prod_{s=1}^m(1+t^{4s-1})$$ when $n=2m+1$, and the form

$$(1+t^{2m-1})\prod_{s=1}^{m-1}(1+t^{4s-1})$$ when $n=2m$. The fundamental group of the manifold $\SO(n)$ is $\Z_2$. The calculation of the higher homotopy groups $\pi_l(\SO(n))$ is directly related to the classification of locally trivial principal $\SO(n)$-fibrations over spheres. An important part in topological $K$-theory is played by the periodicity theorem, according to which, when $N\gg n$, there are the isomorphisms

$$\pi_{n+8}(\O(N)) \simeq \pi_{n}(\O(N));$$ further,

$$\pi_n(\O(N)) \simeq \Z_2$$ if $n=0,1$;

$$\pi_n(\O(N)) \simeq \Z$$ if $n=3,7$; and

$$\pi_n(\O(N)) = 0$$ if $n=2,4,5,6$. The study of the topology of the group $\O(p,q)$ reduces in essence to the previous case, since the connected component of the identity of $\O(p,q)$ is diffeomorphic to the product $\SO(p)\times \SO(q)$ on a Euclidean space.

Comments

A Pythagorean field is a field in which the sum of two squares is again a square.

References

[Ar] E. Artin, "Geometric algebra", Interscience (1957) MR1529733 MR0082463 Zbl 0077.02101
[Bo] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms", 2, Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2272929 MR0928386 MR0896478 MR0782297 MR0782296 MR0722608 MR0682756 MR0643362 MR0647314 MR0610795 MR0583191 MR0354207 MR0360549 MR0237342 MR0205211 MR0205210
[Di] J.A. Dieudonné, "La géométrie des groups classiques", Springer (1955) Zbl 0221.20056
[Di2] J. Dieudonné, "On the automorphisms of the classical groups", Mem. Amer. Math. Soc., 2, Amer. Math. Soc. (1951) MR0045125 Zbl 0042.25603
[Hu] D. Husemoller, "Fibre bundles", McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[OM] O.T. O'Meara, "Introduction to quadratic forms", Springer (1973) Zbl 0259.10018
[We] H. Weyl, "The classical groups, their invariants and representations", Princeton Univ. Press (1946) MR0000255 Zbl 1024.20502
[Zh] D.P. Zhelobenko, "Compact Lie groups and their representations", Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
How to Cite This Entry:
Orthogonal group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Orthogonal_group&oldid=18142
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article