Identity component

In mathematics, the identity component of a topological group G is the connected component G₀ of G that contains the identity element of the group. Similarly, the identity path component of a topological group G is the path component of G that contains the identity element of the group.

Properties

The identity component G₀ of a topological 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 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 = {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 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/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 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, $\pi_0(G,e).$

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.

References

• Lev Semenovich Pontryagin, Topological Groups, 1966.