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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 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 ide ...
''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) 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 g 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 \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
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: ''x''·''y'', or simply ''xy'', ...
s. 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 asJacobson (2009), p. 41 :\mathrm_G(S) = \left\ = \left\, 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 :\mathrm_G(S) = \left\ = \left\, 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 C_G(S) \subseteq N_G(S) and both are subgroups of G.


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 \mathfrak 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 \mathfrak is defined to be :\mathrm_(S) = \. 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) \mathfrak is given by :\mathrm_\mathfrak(S) = \. While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set ''S'' in \mathfrak. If ''S'' is an additive subgroup of \mathfrak, then \mathrm_(S) 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 S' denote the centralizer of S in the semigroup A; i.e. S' = \. Then S' 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 S' = S = S''; 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
by , then we can describe N''G''(''S'') and C''G''(''S'') in terms of the
group action 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 automorphism ...
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 In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring ''S'' of a ring ''R'', denoted C''R''(''S'') in this article. I ...
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 In the branch of abstract algebra called ring theory, the double centralizer theorem can refer to any one of several similar results. These results concern the centralizer of a subring ''S'' of a ring ''R'', denoted C''R''(''S'') in this article. I ...
* 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