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| In [[mathematics]], the '''identity component''' of a [[topological group]] ''G'' is the [[connected component (topology)|connected component]] ''G''<sub>0</sub> 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.
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| == Properties ==
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| The identity component ''G''<sub>0</sub> of a topological group ''G'' is a [[closed set|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 map (topology)|continuous map]]s by definition. Moreover, for any continuous [[automorphism]] ''a'' of ''G'' we have
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| :''a''(''G''<sub>0</sub>) = ''G''<sub>0</sub>. | |
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| Thus, ''G''<sub>0</sub> is a [[characteristic subgroup]] of ''G'', so it is normal.
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| The identity component ''G''<sub>0</sub> of a topological group ''G'' need not be [[open set|open]] in ''G''. In fact, we may have ''G''<sub>0</sub> = {''e''}, in which case ''G'' is [[totally disconnected group|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]].
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| 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.
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| == Component group ==
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| The [[quotient group]] ''G''/''G''<sub>0</sub> is called the '''group of components''' or '''component group''' of ''G''. Its elements are just the connected components of ''G''. The component group ''G''/''G''<sub>0</sub> is a [[discrete group]] if and only if ''G''<sub>0</sub> is open. If ''G'' is an [[affine algebraic group]] then ''G''/''G''<sub>0</sub> is actually a [[finite group]].
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| 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]], <math>\pi_0(G,e).</math>
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| ==Examples==
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| *The group of non-zero real numbers with multiplication ('''R'''*,•) has two components and the group of components is ({1,−1},•).
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| *Consider the [[group of units]] ''U'' in the ring of [[split-complex number]]s. 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''<sub>0</sub> = { ''z'' : |''y''| < ''x'' } . In this case the group of components of ''U'' is isomorphic to the [[Klein four-group]].
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| ==References==
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| *[[Lev Semenovich Pontryagin]], ''Topological Groups'', 1966.
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| [[Category:Topological groups]] | |
| [[Category:Lie groups]]
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