In mathematics, specifically group theory, the identity component of a groupG 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 groupG is the connected componentG0 of G that contains the identity element of the group. The identity path component of a topological groupG is the path component of G that contains the identity element of the group. In algebraic geometry, the identity component of an algebraic groupG over a field k is the identity component of the underlying topological space. The identity component of a group schemeG over a base schemeS is, roughly speaking, the group scheme G0 whose fiber over the point s of S is the connected component 0 of the fiber Gs, an algebraic group.
Properties
The identity component G0 of a topological or algebraic group G is a closednormal 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 automorphisma of G we have Thus, G0 is a characteristic subgroup of G, so it is normal. The identity component G0 of a topological group G need not be open in G. In fact, we may have G0 =, in which case G is totally disconnected. However, the identity component of a locally path-connected space is always open, since it contains a path-connected neighbourhood of ; and therefore is a clopen set. The identity path component of a topological group may in general be smaller than the identity component, but these agree if G is locally path-connected.
Component group
The quotient groupG/G0 is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G0 is a discrete groupif and only ifG0 is open. If G is an algebraic group of finite type, such as an affine algebraic group, then G/G0 is actually a finite group. One may similarly define the path component group as the group of path components, 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 has two components and the group of components is.
Consider the group scheme μ2 = Spec of second roots of unity defined over the base scheme Spec. Topologically, μn consists of two copies of the curve Spec glued together at the point 2. Therefore, μn is connected as a topological space, hence as a scheme. However, μ2does not equal its identity component because the fiber over every point of Spec except 2 consists of two discrete points.
An algebraic group G over a topological fieldK 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 GLn 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 fieldK is totally disconnected in the K-topology and thus has trivial identity component in that topology.
