In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, the centralizer (also called commutant
) of a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''S'' in a
group ''G'' is the set of elements
of ''G'' such that each member
commutes with each element of ''S'', or equivalently, such that
conjugation
Conjugation or conjugate may refer to:
Linguistics
*Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
*Complex conjugation, the change ...
by
leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of elements
of ''G'' that satisfy the weaker condition of leaving the set
fixed under conjugation. The centralizer and normalizer of ''S'' are
subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''.
Suitably formulated, the definitions also apply to
semigroups.
In
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, the centralizer of a subset of a
ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of ''R''. This article also deals with centralizers and normalizers in a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.
The
idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.
Definitions
Group and semigroup
The centralizer of a subset ''S'' of group (or semigroup) ''G'' is defined as
[Jacobson (2009), p. 41]
:
where only the first definition applies to semigroups.
If there is no ambiguity about the group in question, the ''G'' can be suppressed from the notation. When ''S'' = is a
singleton set, we write C
''G''(''a'') instead of C
''G''(). Another less common notation for the centralizer is Z(''a''), which parallels the notation for the
center. With this latter notation, one must be careful to avoid confusion between the center of a group ''G'', Z(''G''), and the ''centralizer'' of an ''element'' ''g'' in ''G'', Z(''g'').
The normalizer of ''S'' in the group (or semigroup) ''G'' is defined as
:
where again only the first definition applies to semigroups. The definitions are similar but not identical. If ''g'' is in the centralizer of ''S'' and ''s'' is in ''S'', then it must be that , but if ''g'' is in the normalizer, then for some ''t'' in ''S'', with ''t'' possibly different from ''s''. That is, elements of the centralizer of ''S'' must commute pointwise with ''S'', but elements of the normalizer of ''S'' need only commute with ''S as a set''. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the
normal closure.
Clearly
and both are subgroups of
.
Ring, algebra over a field, Lie ring, and Lie algebra
If ''R'' is a ring or an
algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
, and ''S'' is a subset of ''R'', then the centralizer of ''S'' is exactly as defined for groups, with ''R'' in the place of ''G''.
If
is a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
(or
Lie ring) with Lie product
'x'', ''y'' then the centralizer of a subset ''S'' of
is defined to be
:
The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If ''R'' is an associative ring, then ''R'' can be given the
bracket product . Of course then if and only if . If we denote the set ''R'' with the bracket product as L
''R'', then clearly the ''ring centralizer'' of ''S'' in ''R'' is equal to the ''Lie ring centralizer'' of ''S'' in L
''R''.
The normalizer of a subset ''S'' of a Lie algebra (or Lie ring)
is given by
:
While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the
idealizer of the set ''S'' in
. If ''S'' is an additive subgroup of
, then
is the largest Lie subring (or Lie subalgebra, as the case may be) in which ''S'' is a Lie
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
.
Properties
Semigroups
Let
denote the centralizer of
in the semigroup
; i.e.
Then
forms a
subsemigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
and
; i.e. a commutant is its own
bicommutant In algebra, the bicommutant of a subset ''S'' of a semigroup (such as an algebra or a group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant and is written S^.
The bicommutant is part ...
.
Groups
Source:
* The centralizer and normalizer of ''S'' are both subgroups of ''G''.
* Clearly, . In fact, C
''G''(''S'') is always a
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 N
''G''(''S''), being the kernel of the homomorphism and the group N
''G''(''S'')/C
''G''(''S'') acts by conjugation as a group of bijections on ''S''. E.g. 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 compact
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 addi ...
''G'' with a torus ''T'' is defined as , and especially if the torus is maximal (i.e. it is a central tool in the theory of Lie groups.
* C
''G''(C
''G''(''S'')) contains ''S'', but C
''G''(''S'') need not contain ''S''. Containment occurs exactly when ''S'' is abelian.
* If ''H'' is a subgroup of ''G'', then N
''G''(''H'') contains ''H''.
* If ''H'' is a subgroup of ''G'', then the largest subgroup of ''G'' in which ''H'' is normal is the subgroup N
''G''(H).
* If ''S'' is a subset of ''G'' such that all elements of ''S'' commute with each other, then the largest subgroup of ''G'' whose center contains ''S'' is the subgroup C
''G''(S).
* A subgroup ''H'' of a group ''G'' is called a of ''G'' if .
* The center of ''G'' is exactly C
''G''(G) and ''G'' is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
if and only if .
* For singleton sets, .
* By symmetry, if ''S'' and ''T'' are two subsets of ''G'', if and only if .
* For a subgroup ''H'' of group ''G'', the N/C theorem states that the
factor group
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, s ...
N
''G''(''H'')/C
''G''(''H'') is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to a subgroup of Aut(''H''), the group of
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 automorphis ...
s of ''H''. Since and , the N/C theorem also implies that ''G''/Z(''G'') is isomorphic to Inn(''G''), the subgroup of Aut(''G'') consisting of all
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
s of ''G''.
* If we define a
group homomorphism by , then we can describe N
''G''(''S'') and C
''G''(''S'') in terms of the
group action of Inn(''G'') on ''G'': the stabilizer of ''S'' in Inn(''G'') is ''T''(N
''G''(''S'')), and the subgroup of Inn(''G'') fixing ''S'' pointwise is ''T''(C
''G''(''S'')).
* A subgroup ''H'' of a group ''G'' is said to be C-closed or self-bicommutant if for some subset . If so, then in fact, .
Rings and algebras over a field
Source:
* Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively.
* The normalizer of ''S'' in a Lie ring contains the centralizer of ''S''.
* C
''R''(C
''R''(''S'')) contains ''S'' but is not necessarily equal. The
double centralizer theorem deals with situations where equality occurs.
* If ''S'' is an additive subgroup of a Lie ring ''A'', then N
''A''(''S'') is the largest Lie subring of ''A'' in which ''S'' is a Lie ideal.
* If ''S'' is a Lie subring of a Lie ring ''A'', then .
See also
*
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
*
Double centralizer theorem
*
Idealizer
*
Multipliers and centralizers (Banach spaces)
*
Stabilizer subgroup
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
Notes
References
*
*
*
{{DEFAULTSORT:Centralizer And Normalizer
Abstract algebra
Group theory
Ring theory
Lie algebras