# Identity component

In mathematics, specifically group theory, the **identity component** of a group *G* refers to several closely related notions of the largest connected subgroup of *G* containing the identity element.

In point set topology, the **identity component of a topological group** *G* is the connected component *G*^{0} of *G* that contains the identity element of the group. The **identity path component of a topological group** *G* is the path component of *G* that contains the identity element of the group.

In algebraic geometry, the **identity component of an algebraic group** *G* over a field *k* is the identity component of the underlying topological space. The **identity component of a group scheme** *G* over a base scheme *S* is, roughly speaking, the group scheme *G*^{0} whose fiber over the point *s* of *S* is the connected component *(G _{s})^{0}* of the fiber

*G*, an algebraic group.[1]

_{s}## Properties

The identity component *G*^{0} of a topological or algebraic group *G* is a closed normal subgroup of *G*. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological or algebraic group are continuous maps by definition. Moreover, for any continuous automorphism *a* of *G* we have

*a*(*G*^{0}) =*G*^{0}.

Thus, *G*^{0} is a characteristic subgroup of *G*, so it is normal.

The identity component *G*^{0} of a topological group *G* need not be open in *G*. In fact, we may have *G*^{0} = {*e*}, in which case *G* is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {*e*}; and therefore is a clopen set.

The identity path component of a topological group may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if *G* is locally path-connected.

## Component group

The quotient group *G*/*G*^{0} is called the **group of components** or **component group** of *G*. Its elements are just the connected components of *G*. The component group *G*/*G*^{0} is a discrete group if and only if *G*^{0} is open. If *G* is an algebraic group of finite type, such as an affine algebraic group, then *G*/*G*^{0} is actually a finite group.

One may similarly define the path component group as the group of path components (quotient of *G* by the identity path component), and in general the component group is a quotient of the path component group, but if *G* is locally path connected these groups agree. The path component group can also be characterized as the zeroth homotopy group,

## Examples

- The group of non-zero real numbers with multiplication (
**R***,•) has two components and the group of components is ({1,−1},•). - Consider the group of units
*U*in the ring of split-complex numbers. In the ordinary topology of the plane {*z*=*x*+ j*y*:*x*,*y*∈**R**},*U*is divided into four components by the lines*y*=*x*and*y*= −*x*where*z*has no inverse. Then*U*^{0}= {*z*: |*y*| <*x*} . In this case the group of components of*U*is isomorphic to the Klein four-group. - The identity component of the additive group (
**Z**_{p},+) of p-adic integers is the singleton set {1}, since**Z**_{p}is totally disconnected. - The Weyl group of a reductive algebraic group
*G*is the components group of the normalizer group of a maximal torus of*G*. - Consider the group scheme μ
_{2}= Spec(**Z**[*x*]/(*x*^{2}- 1)) of second roots of unity defined over the base scheme Spec(**Z**). Topologically, μ_{n}consists of two copies of the curve Spec(**Z**) glued together at the point (that is, prime ideal) 2. Therefore, μ_{n}is connected as a topological space, hence as a scheme. However, μ_{2}does not equal its identity component because the fiber over every point of Spec(**Z**) except 2 consists of two discrete points.

An algebraic group *G* over a topological field *K* admits two natural topologies, the Zariski topology and the topology inherited from *K*. The identity component of *G* often changes depending on the topology. For instance, the general linear group GL_{n}(**R**) is connected as an algebraic group but has two path components as a Lie group, the matrices of positive determinant and the matrices of negative determinant. Any connected algebraic group over a non-Archimedean local field *K* is totally disconnected in the *K*-topology and thus has trivial identity component in that topology.

## References

- SGA 3, v. 1, Exposé VI, Définition 3.1

- Lev Semenovich Pontryagin,
*Topological Groups*, 1966. - Demazure, Michel; Gabriel, Pierre (1970),
*Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs*, Paris: Masson, ISBN 978-2225616662, MR 0302656