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''identity component, of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251001.png" />''
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''identity component, of a group $G$''
  
The largest connected subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251002.png" /> of the topological (or algebraic) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251003.png" /> that contains the identity element of this group. The component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251004.png" /> is a closed normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251005.png" />; the cosets with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251006.png" /> coincide with the connected components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251007.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251008.png" /> is totally disconnected and Hausdorff, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251009.png" /> is the smallest among the normal subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510011.png" /> is totally disconnected. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510012.png" /> is locally connected (for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510013.png" /> is a Lie group), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510014.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510016.png" /> is discrete.
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The largest connected subset $G^0$ of the topological (or algebraic) group $G$ that contains the identity element of this group. The component $G^0$ is a closed normal subgroup of $G$; the cosets with respect to $G^0$ coincide with the connected components of $G$. The quotient group $G/G^0$ is totally disconnected and Hausdorff, and $G^0$ is the smallest among the normal subgroups $H\subset G$ such that $G/H$ is totally disconnected. If $G$ is locally connected (for example, if $G$ is a Lie group), then $G^0$ is open in $G$ and $G/G^0$ is discrete.
  
In an arbitrary algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510017.png" /> the identity component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510018.png" /> is also open and has finite index; also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510019.png" /> is the minimal closed subgroup of finite index in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510020.png" />. The connected components of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510021.png" /> coincide with the irreducible components. For every polynomial homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510022.png" /> of algebraic groups one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510024.png" /> is defined over a field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510025.png" /> is defined over this field.
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In an arbitrary algebraic group $G$ the identity component $G^0$ is also open and has finite index; also, $G^0$ is the minimal closed subgroup of finite index in $G$. The connected components of an algebraic group $G$ coincide with the irreducible components. For every polynomial homomorphism $\phi$ of algebraic groups one has $\phi(G^0)=\phi(G)^0$. If $G$ is defined over a field, then $G^0$ is defined over this field.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510026.png" /> is an algebraic group over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510027.png" />, then its identity component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510028.png" /> coincides with the identity component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510029.png" /> considered as a complex Lie group. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510030.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510031.png" />, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510032.png" /> of real points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510033.png" /> is not necessarily connected in the topology of the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510034.png" />, but the number of its connected components is finite. For example, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510035.png" /> splits into two components, although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510036.png" /> is connected. The pseudo-orthogonal unimodular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510037.png" />, which can be regarded as the group of real points of the connected complex algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510038.png" />, is connected for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510039.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510040.png" />, and splits into two components for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510041.png" />. However, if the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510042.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c02510043.png" /> is connected.
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If $G$ is an algebraic group over the field $\mathbf C$, then its identity component $G^0$ coincides with the identity component of $G$ considered as a complex Lie group. If $G$ is defined over $\mathbf R$, then the group $G^0(\mathbf R)$ of real points in $G^0$ is not necessarily connected in the topology of the Lie group $G(\mathbf R)$, but the number of its connected components is finite. For example, the group $\operatorname{GL}_n(\mathbf R)$ splits into two components, although $\operatorname{GL}_n(\mathbf C)$ is connected. The pseudo-orthogonal unimodular group $\operatorname{SO}(p,q)$, which can be regarded as the group of real points of the connected complex algebraic group $\operatorname{SO}_{p+q}(\mathbf C)$, is connected for $p=0$ or $q=0$, and splits into two components for $p,q>0$. However, if the Lie group $G(\mathbf R)$ is compact, then $G^0(\mathbf R)$ is connected.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)  {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.R. Shafarevich,  "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)  {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)  {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,  "Differential geometry and symmetric spaces" , Acad. Press  (1962)  {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.R. Shafarevich,  "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)  {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>

Revision as of 15:51, 17 July 2014

identity component, of a group $G$

The largest connected subset $G^0$ of the topological (or algebraic) group $G$ that contains the identity element of this group. The component $G^0$ is a closed normal subgroup of $G$; the cosets with respect to $G^0$ coincide with the connected components of $G$. The quotient group $G/G^0$ is totally disconnected and Hausdorff, and $G^0$ is the smallest among the normal subgroups $H\subset G$ such that $G/H$ is totally disconnected. If $G$ is locally connected (for example, if $G$ is a Lie group), then $G^0$ is open in $G$ and $G/G^0$ is discrete.

In an arbitrary algebraic group $G$ the identity component $G^0$ is also open and has finite index; also, $G^0$ is the minimal closed subgroup of finite index in $G$. The connected components of an algebraic group $G$ coincide with the irreducible components. For every polynomial homomorphism $\phi$ of algebraic groups one has $\phi(G^0)=\phi(G)^0$. If $G$ is defined over a field, then $G^0$ is defined over this field.

If $G$ is an algebraic group over the field $\mathbf C$, then its identity component $G^0$ coincides with the identity component of $G$ considered as a complex Lie group. If $G$ is defined over $\mathbf R$, then the group $G^0(\mathbf R)$ of real points in $G^0$ is not necessarily connected in the topology of the Lie group $G(\mathbf R)$, but the number of its connected components is finite. For example, the group $\operatorname{GL}_n(\mathbf R)$ splits into two components, although $\operatorname{GL}_n(\mathbf C)$ is connected. The pseudo-orthogonal unimodular group $\operatorname{SO}(p,q)$, which can be regarded as the group of real points of the connected complex algebraic group $\operatorname{SO}_{p+q}(\mathbf C)$, is connected for $p=0$ or $q=0$, and splits into two components for $p,q>0$. However, if the Lie group $G(\mathbf R)$ is compact, then $G^0(\mathbf R)$ is connected.

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[3] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101
[4] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
How to Cite This Entry:
Connected component of the identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Connected_component_of_the_identity&oldid=32483
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article