# Difference between revisions of "Connected component of the identity"

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− | <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> | + | <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 10:02, 24 March 2012

*identity component, of a group *

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

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

If is an algebraic group over the field , then its identity component coincides with the identity component of considered as a complex Lie group. If is defined over , then the group of real points in is not necessarily connected in the topology of the Lie group , but the number of its connected components is finite. For example, the group splits into two components, although is connected. The pseudo-orthogonal unimodular group , which can be regarded as the group of real points of the connected complex algebraic group , is connected for or , and splits into two components for . However, if the Lie group is compact, then 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=19121