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 ...
, especially
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 centralizer (also called commutant
) of a
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
''S'' in a
group ''G'' is the set
of elements of ''G'' that
commute with every element of ''S'', or equivalently, the set of elements
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 o ...
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
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
s 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
semigroup
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 (just notation, not necessarily th ...
s.
In
ring theory, the centralizer of a subset of a
ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring ''R'' is a
subring
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...
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 ident ...
.
The
idealizer In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by
:\mathbb_S(T)=\.
In ring theory, if ...
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 ''
'' 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
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. If the set
is a subgroup of
, then the normalizer
is the largest subgroup
where
is 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 ...
of
. The definitions of ''centralizer'' and ''normalizer'' are similar but not identical. If ''g'' is in the centralizer of ''
'' and ''s'' is in ''
'', then it must be that , but if ''g'' is in the normalizer, then for some ''t'' in ''
'', with ''t'' possibly different from ''s''. That is, elements of the centralizer of ''
'' must commute pointwise with ''
'', 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 map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
, and ''
'' is a subset of ''R'', then the centralizer of ''
'' 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 ident ...
(or
Lie ring
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 ...
) with Lie product
'x'', ''y'' then the centralizer of a subset ''
'' 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 ''
'' in ''R'' is equal to the ''Lie ring centralizer'' of ''
'' in L
''R''.
The normalizer of a subset ''
'' 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 In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by
:\mathbb_S(T)=\.
In ring theory, if ...
of the set ''
'' in
. If ''
'' is an additive subgroup of
, then
is the largest Lie subring (or Lie subalgebra, as the case may be) in which ''
'' is a Lie
ideal.
Example
Consider the group
:
(the symmetric group of permutations of 3 elements).
Take a subset
of the group
:
:
Note that
3 is
H itself (
, 2, 3 , 3, 2.
Properties
Semigroups
Let
S' denote the centralizer of
S in the semigroup
A; i.e.
S' = \. Then
S' forms a
subsemigroup and
S' = S = S''; i.e. a commutant is its own
bicommutant.
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 ...
of N
''G''(''S''), being the kernel of the
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
and the group N
''G''(''S'')/C
''G''(''S'')
acts by conjugation as a
group of bijections on ''S''. E.g. the
Weyl group of a compact
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 ...
''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 commu ...
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 N
''G''(''H'')/C
''G''(''H'') is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
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 automorphism ...
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 automorphisms 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, ...
*
Multipliers and centralizers (Banach spaces)
*
Stabilizer subgroup
Notes
References
*
*
*
{{DEFAULTSORT:Centralizer And Normalizer
Abstract algebra
Group theory
Ring theory
Lie algebras