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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, the identity component of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
''G'' (also known as its unity component) refers to several closely related notions of the largest
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
subgroup of ''G'' containing the identity element. In
point set topology In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differ ...
, the identity component of a
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
''G'' is the connected component ''G''0 of ''G'' that contains the
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of the group. The identity path component of a topological group ''G'' is the
path component In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties th ...
of ''G'' that contains the identity element of the group. In
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, the identity component of an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
''G'' over a field ''k'' is the identity component of the underlying topological space. The identity component of a
group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
''G'' over a base scheme ''S'' is, roughly speaking, the group scheme ''G''0 whose
fiber Fiber (spelled fibre in British English; from ) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often inco ...
over the point ''s'' of ''S'' is the connected component ''G''''s''0 of the fiber ''Gs'', an algebraic group.SGA 3, v. 1, Exposé VIB, Définition 3.1


Properties

The identity component ''G''0 of a topological or algebraic group ''G'' is a closed
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
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 In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s by definition. Moreover, for any continuous
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
''a'' of ''G'' we have :''a''(''G''0) = ''G''0. Thus, ''G''0 is a 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 Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979 * ''Open'' (Go ...
in ''G''. In fact, we may have ''G''0 = , in which case ''G'' is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
. However, the identity component of a
locally path-connected space In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting of open connected sets. As a stronger notion, the space ''X'' is locally path connected if e ...
(for instance a
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
) is always open, since it contains a
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
neighbourhood of ; and therefore is a
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical de ...
.


Component group

The
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
''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 In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
if and only if ''G''0 is open. If ''G'' is an algebraic group of finite type, such as an
affine algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
, then ''G''/''G''0 is actually a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
. 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 In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
, \pi_0(G,e).


Examples

*The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is (,•). *Consider the
group of units In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the ele ...
''U'' in the ring of
split-complex number In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying j^2=1, where j \neq \pm 1. A split-complex number has two real number components and , and is written z=x+y ...
s. In the ordinary topology of the plane , ''U'' is divided into four components by the lines ''y'' = ''x'' and ''y'' = − ''x'' where ''z'' has no inverse. Then ''U''0 = . In this case the group of components of ''U'' is isomorphic to the
Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
. *The identity component of the additive group (Zp,+) of p-adic integers is the singleton set , since Zp is totally disconnected. *The
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
of a reductive algebraic group ''G'' is the components group of the normalizer group of a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
of ''G''. *Consider the group scheme μ''2'' = Spec(Z 'x''(''x''2 - 1)) of second 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 In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
) 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 field ''K'' admits two natural topologies, the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
and the topology inherited from ''K''. The identity component of ''G'' often changes depending on the topology. For instance, the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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 In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
''K'' is totally disconnected in the ''K''-topology and thus has trivial identity component in that topology.


note


References

* Lev Semenovich Pontryagin, ''Topological Groups'', 1966. * *


External links

* Revised and annotated edition of the 1970 original. {{DEFAULTSORT:Identity component Topological groups Lie groups