# 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.

_{s}^{[1]}

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

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.

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.

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.