Connected component of the identity

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