In
mathematics, 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 ''G'' refers to several closely related notions of the largest
connected subgroup of ''G'' containing the identity element.
In
point set topology, the identity component of a
topological group
In mathematics, topological groups are logically 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 ...
''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 operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
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 ...
of ''G'' that contains the identity element of the group.
In
algebraic geometry, the identity component of an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
...
''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 ha ...
''G'' over a base
scheme ''S'' is, roughly speaking, the group scheme ''G''
0 whose
fiber
Fiber or fibre (from la, fibra, links=no) 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 incorpora ...
over the point ''s'' of ''S'' is the connected component ''(G
s)
0'' of the fiber ''G
s'', an algebraic group.
[SGA 3, v. 1, Exposé VI, 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 G ...
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 continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
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
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
of ''G'', so it is normal.
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'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (Y ...
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 (for instance a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
) 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 ...
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 counter-intuitive, as the common meanings of and are antonyms, but their mathematical d ...
.
The identity path component of a topological group 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.
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 exam ...
''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 o ...
if and only if ''G''
0 is open. If ''G'' is an algebraic group of
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
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, homot ...
,
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 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 element is unique for th ...
''U'' in the ring of
split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
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 a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third on ...
.
*The identity component of the additive group (Z
p,+) of
p-adic integers is the singleton set , since Z
p 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 ...
of a
reductive algebraic group ''G'' is the components group of the
normalizer group of a
maximal torus 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) 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 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 invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible ...
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 topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...
''K'' is totally disconnected in the ''K''-topology and thus has trivial identity component in that topology.
note
References
*
Lev Semenovich Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
, ''Topological Groups'', 1966.
*
*
External links
* Revised and annotated edition of the 1970 original.
{{DEFAULTSORT:Identity component
Topological groups
Lie groups