Identity component

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

The identity component G₀ is a closed, normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion are continuous maps. Moreover, for any continuous automorphism a of G we have

a(G₀) = G₀.

It follows that G₀ is normal in G.

It is not always true that G₀ is open in G. In fact, we may have G₀ = {e}, in which case G is totally disconnected. However, if G is a Lie group then G₀ is open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set. More generally, for any locally connected topological group the identity component G₀ is clopen.

The quotient group G/G₀ is called the group of components 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.

[edit] 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.

[edit] References

• Lev Semenovich Pontryagin, Topological Groups, 1966.