In mathematics, specifically group theory, the identity component of a group G (also known as its unity component) 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 G0 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 G0 whose fiber over the point s of S is the connected component (Gs)0 of the fiber Gs, an algebraic group.[1]
The identity component G0 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(G0) = G0.
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 (for instance a Lie group) 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 (since path connectedness is a stronger condition than connectedness), but these agree if G is locally path-connected.
The quotient group G/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 group if and only if G0 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 (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,
\pi0(G,e).
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 GLn(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.