Identity component

2025-02-15T05:50:11Z

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{{Short description|Concept in group theory}}
{{no footnotes|date=June 2016}}

In [[mathematics]], specifically [[group theory]], the '''identity component''' of a [[group (mathematics) |group]] ''G'' (also known as its '''unity component''') refers to several closely related notions of the largest [[connected space |connected]] subgroup of ''G'' containing the identity element.

In [[point set topology]], the '''identity component of a [[topological group]]''' ''G'' is the [[connected component (topology)|connected component]] ''G''0 of ''G'' that contains the [[identity element]] of the group. The '''identity path component of a topological group''' ''G'' is the [[path component]] of ''G'' that contains the identity element of the group.

In [[algebraic geometry]], the '''identity component of an [[algebraic group]]''' ''G'' over a field ''k'' is the identity component of the underlying topological space. The '''identity component of a [[group scheme]]''' ''G'' over a base [[scheme (mathematics) |scheme]] ''S'' is, roughly speaking, the group scheme ''G''0 whose [[fiber (mathematics) |fiber]] over the point ''s'' of ''S'' is the connected component ''G''''s''0 of the fiber ''Gs'', an algebraic group.<ref>SGA 3, v. 1, Exposé VIB, Définition 3.1</ref>

== Properties ==
The identity component ''G''0 of a topological or algebraic 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 or algebraic group are [[continuous map (topology)|continuous map]]s by definition. Moreover, for any continuous [[automorphism]] ''a'' of ''G'' we have

:''a''(''G''0) = ''G''0.

Thus, ''G''0 is a [[characteristic subgroup|characteristic]] (topological or algebraic) subgroup of ''G'', so it is normal.

By the same argument as above, the identity path component of a topological group is also a normal subgroup (characteristic as a topological subgroup). It 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.

The identity component ''G''0 of a topological group ''G'' need not be [[open set|open]] in ''G''. In fact, we may have ''G''0 = {''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]].

== Component group ==
The [[quotient group]] ''G''/''G''0 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''0 is a [[discrete group]] if and only if ''G''0 is open. If ''G'' is an algebraic group of [[glossary of algebraic geometry | finite type]], such as an [[affine algebraic group]], then ''G''/''G''0 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]], <math>\pi_0(G,e).</math>

==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 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''0 = { ''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 ('''Z'''p,+) of [[p-adic number | p-adic integers]] is the singleton set {0}, since '''Z'''p is totally disconnected.
*The [[Weyl group]] of a [[reductive group | reductive algebraic group]] ''G'' is the components group of the [[centralizer and normalizer | normalizer group]] of a [[maximal torus]] of ''G''.
*Consider the group scheme μ''2'' = Spec('''Z'''[''x'']/(''x''2 - 1)) of second [[root of unity | roots of unity]] defined over the base scheme Spec('''Z'''). Topologically, μ''n'' consists of two copies of the curve Spec('''Z''') glued together at the point (that is, [[prime ideal]]) 2. Therefore, μ''n'' is connected as a topological space, hence as a scheme. However, μ''2'' 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 ring | 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]] GL''n''('''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 ==
{{reflist}}
==References==
{{ref begin}}
*[[Lev Semenovich Pontryagin]], ''Topological Groups'', 1966.
*{{Citation | author1-last=Demazure | author1-first=Michel | author1-link=Michel Demazure | author2-last=Gabriel | author2-first=Pierre | author2-link=Pierre Gabriel | title=Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs | publisher=Masson | location=Paris | year=1970 | isbn=978-2225616662 | mr=0302656}}
*{{cite book |editor-last = Demazure |editor-first = Michel |editor-link = Michel Demazure |editor2=Alexandre Grothendieck |editor2-link=Alexandre Grothendieck | title = Propriétés Générales des Schémas en Groupes |series = Lecture Notes in Mathematics | year = 1970 |volume = 151| publisher = [[Springer Science+Business Media|Springer-Verlag]] | location = Berlin; New York | language = fr | pages = xv+564|doi=10.1007/BFb0058993|isbn=978-3-540-05179-4 | mr = 0274458 }}
{{ref end}}
==External links==
*{{Citation | author1-last=Demazure | author1-first=M. | author1-link=Michel Demazure | author2-last=Grothendieck | author2-first=A. | author2-link=Alexander Grothendieck | editor1-last=Gille | editor1-first=P. | editor2-last=Polo | editor2-first=P. | title = Schémas en groupes (SGA 3), I: Propriétés Générales des Schémas en Groupes | url=https://webusers.imj-prg.fr/~patrick.polo/SGA3/}} Revised and annotated edition of the 1970 original.

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[[Category:Topological groups]]
[[Category:Lie groups]]

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Identity component