IDENTITY COMPONENT

In mathematics, the 'identity component' of a topological group ''G'' is the connected component ''G''₀ that contains the identity element ''e''.
The identity component ''G''₀ 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''(''G''₀) = ''G''₀.
It follows that ''G''₀ is normal in ''G''.
It is not always true that ''G''₀ is open in ''G''. In fact, we may have ''G''₀ = {''e''}, in which case ''G'' is totally disconnected. However, if ''G'' is a Lie group then ''G''₀ 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 ''G''₀ is clopen.
The quotient group ''G''/''G''₀ is called the 'group of components' of ''G''. Its elements are just the connected components of ''G''. The component group ''G''/''G''₀ is a discrete group if and only if ''G''₀ is open. If ''G'' is an affine algebraic group then ''G''/''G''₀ is actually a finite group.

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Examples

Reference

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 ''U''₀ = { ''z'' : |''y''| < ''x'' } . In this case the group of components of ''U'' is isomorphic to the Klein four-group.

Reference

★ Lev Semenovich Pontryagin, Topological Groups, 1966.