View source for Connected component of the identity

''identity component, of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025100/c0251001.png" />''

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.

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.

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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Helgason,   "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.R. Shafarevich,   "Basic algebraic geometry" , Springer  (1977)  (Translated from Russian)</TD></TR></table>