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 U0 = { 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 (Zp,+) of p-adic integers is the singleton set {0}, since Zp 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 μ2 = Spec(Z[x]/(x2 - 1)) of second roots of unity defined over the base scheme Spec(Z). Topologically, μn consists of two copies of the curve Spec(Z) glued together at the point (that is, prime ideal) 2. Therefore, μn is connected as a topological space, hence as a scheme. However, μ2 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 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. note References Lev Semenovich Pontryagin, Topological Groups, 1966. External links Revised and annotated edition of the 1970 original. Topological groups Lie groups