Identity component

<h2 id="properties">Properties</h2>
The identity component G0 of a topological or algebraic group G is a <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> 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 <a href="/facts/Continuous_map_(topology)/sKbl02pB">continuous maps</a> by definition. Moreover, for any continuous <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> a of G we have 

a(G0) = G0.
Thus, G0 is a <a href="/facts/Characteristic_subgroup/UfoS7U7S">characteristic</a> (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 G0 of a topological group G need not be <a href="/facts/Open_set/QfTCIqMu">open</a> in G. In fact, we may have G0 = {e}, in which case G is <a href="/facts/Totally_disconnected_group/7LbMsa8N">totally disconnected</a>. However, the identity component of a <a href="/facts/Locally_path-connected_space/5CUTxfoA">locally path-connected space</a> (for instance a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>) is always open, since it contains a <a href="/facts/Path-connected/5CUTxfoA">path-connected</a> neighbourhood of {e}; and therefore is a <a href="/facts/Clopen_set/7Vrf6PYs">clopen set</a>.

<h2 id="component-group">Component group</h2>
The <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> 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 <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a> if and only if G0 is open. If G is an algebraic group of <a href="/facts/Glossary_of_algebraic_geometry/eh7NLhmS"> finite type</a>, such as an <a href="/facts/Affine_algebraic_group/eMLtu5Nh">affine algebraic group</a>, then G/G0 is actually a <a href="/facts/Finite_group/vv2tvogB">finite group</a>.
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 <a href="/facts/Homotopy_group/xjogL95Y">homotopy group</a>, 
 
 
 
 
 π
 
 0
 
 
 (
 G
 ,
 e
 )
 .
 
 
 {\displaystyle \pi _{0}(G,e).}

<h2 id="examples">Examples</h2>
<ul><li>The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is ({1,−1},•).</li>
<li>Consider the <a href="/facts/Group_of_units/GJSBu8Y7">group of units</a> U in the ring of <a href="/facts/Split-complex_number/DtavqChA">split-complex numbers</a>. 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 <a href="/facts/Klein_four-group/X2oQCYBy">Klein four-group</a>.</li>
<li>The identity component of the additive group (Zp,+) of <a href="/facts/P-adic_number/SDgDbkLF"> p-adic integers</a> is the singleton set {0}, since Zp is totally disconnected.</li>
<li>The <a href="/facts/Weyl_group/N9QG4Wyn">Weyl group</a> of a <a href="/facts/Reductive_group/z1zyJvUh"> reductive algebraic group</a> G is the components group of the <a href="/facts/Centralizer_and_normalizer/AKhrEUGR"> normalizer group</a> of a <a href="/facts/Maximal_torus/f82Qln74">maximal torus</a> of G.</li>
<li>Consider the group scheme μ2 = Spec(Z[x]/(x2 - 1)) of second <a href="/facts/Root_of_unity/EqH0ho1Y"> roots of unity</a> defined over the base scheme Spec(Z). Topologically, μn consists of two copies of the curve Spec(Z) glued together at the point (that is, <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a>) 2. Therefore, μn is connected as a topological space, hence as a scheme. However, μ2 does not equal its identity component because the fiber over every point of Spec(Z) except 2 consists of two discrete points.</li></ul>
An algebraic group G over a <a href="/facts/Topological_ring/qxQqzhCy"> topological field</a> K admits two natural topologies, the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a> and the topology inherited from K. The identity component of G often changes depending on the topology. For instance, the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> GLn(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 <a href="/facts/Local_field/vqIgPIX3">local field</a> K is totally disconnected in the K-topology and thus has trivial identity component in that topology.

<h2 id="note">note</h2>

<ul><li><a href="/facts/Lev_Semenovich_Pontryagin/fDdDv9rA">Lev Semenovich Pontryagin</a>, Topological Groups, 1966.</li>
<li><a href="/facts/Michel_Demazure/vEzljFW7">Demazure, Michel</a>; <a href="/facts/Pierre_Gabriel/g95YjYqI">Gabriel, Pierre</a> (1970), Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Paris: Masson, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-2225616662, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0302656">0302656</a></li>
<li><a href="/facts/Michel_Demazure/vEzljFW7">Demazure, Michel</a>; <a href="/facts/Alexandre_Grothendieck/jLQGPY7M">Alexandre Grothendieck</a>, eds. (1970). Propriétés Générales des Schémas en Groupes. Lecture Notes in Mathematics (in French). Vol. 151. Berlin; New York: <a href="/facts/Springer_Science%252BBusiness_Media/nAesf6nT">Springer-Verlag</a>. pp. xv+564. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0058993">10.1007/BFb0058993</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05179-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0274458">0274458</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Michel_Demazure/vEzljFW7">Demazure, M.</a>; <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Grothendieck, A.</a>, Gille, P.; Polo, P. (eds.), <a href="https://webusers.imj-prg.fr/~patrick.polo/SGA3/">Schémas en groupes (SGA 3), I: Propriétés Générales des Schémas en Groupes</a> Revised and annotated edition of the 1970 original.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">SGA 3, v. 1, Exposé VIB, Définition 3.1 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Identity component open-in-new

Identity component