TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, especially
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, the centralizer (also called commutant) of a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... ''S'' in a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''G'' is the set of elements $C_G\left(S\right)$ of ''G'' such that each member $g \in C_G\left(S\right)$
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 of elements $N_G\left(S\right)$ 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
s and
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
s. In
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of ''R''. This article also deals with centralizers and normalizers in a
Lie algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. The
idealizer In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
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 :$\mathrm_G\left(S\right) = \left\ = \left\.$ 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 Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
. 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 :$\mathrm_G\left(S\right) = \left\ = \left\.$ 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 $C_G\left(S\right) \subseteq N_G\left(S\right)$ 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 map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
(or
Lie ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
) with Lie product 'x'', ''y'' then the centralizer of a subset ''S'' of $\mathfrak$ is defined to be :$\mathrm_\left(S\right) = \.$ 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 :$\mathrm_\mathfrak\left(S\right) = \.$ While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the
idealizer In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
of the set ''S'' in $\mathfrak$. If ''S'' is an additive subgroup of $\mathfrak$, then $\mathrm_\left(S\right)$ 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) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
.

# Properties

## Semigroups

Let $S\text{'}$ denote the centralizer of $S$ in the semigroup $A$; i.e. $S\text{'} = \.$ Then $S\text{'}$ forms a
subsemigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation. The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...
and $S\text{'} = S = S\text{'}\text{'}$; i.e. a commutant is its own
bicommutant In algebra, the bicommutant of a subset ''S'' of a semigroup (such as an algebra over a field, algebra or a group (mathematics), group) is the commutant of the commutant of that subset. It is also known as the double commutant or second commutant an ...
.

## Groups

Source: * The centralizer and normalizer of ''S'' are both subgroups of ''G''. * Clearly, $C_G\left(S\right) \subseteq N_G\left(S\right)$. In fact, $C_G\left(S\right)$ is always a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of $N_G\left(S\right)$, being the kernel of the homomorphism $N_G\left(S\right)\to \operatorname\left(S\right)$ and the group $N_G\left(S\right)/C_G\left(S\right)$ acts by conjugation as a group of bijections on $S$ . E.g. the
Weyl group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of a compact
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
$G$ with a torus $T$ is defined as $W\left(G, H\right) = N_G\left(T\right)/C_G\left(T\right)$, and especially if the torus is maximal (i.e. $C_G\left(T\right) = T$) 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 N''G''(''H'') = ''H''. * The center of ''G'' is exactly C''G''(G) and ''G'' is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
if and only if C''G''(G) = Z(''G'') = ''G''. * For singleton sets, C''G''(''a'') = N''G''(''a''). * By symmetry, if ''S'' and ''T'' are two subsets of ''G'', ''T'' ⊆ C''G''(''S'') if and only if ''S'' ⊆ C''G''(''T''). * For a subgroup ''H'' of group ''G'', the N/C theorem states that the
factor group A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
N''G''(''H'')/C''G''(''H'') is
isomorphic In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
to a subgroup of Aut(''H''), the group of
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... s of ''H''. Since N''G''(''G'') = ''G'' and C''G''(''G'') = Z(''G''), 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s of ''G''. * If we define a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ... ''T'' : ''G'' → Inn(''G'') by ''T''(''x'')(''g'') = ''T''''x''(''g'') = ''xgx''−1, then we can describe N''G''(''S'') and C''G''(''S'') in terms of the
group action In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 ''H'' = C''G''(''S'') for some subset ''S'' ⊆ ''G''. If so, then in fact, ''H'' = C''G''(C''G''(''H'')).

## 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (m ...
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 ''S'' ⊆ N''A''(''S'').

*
Commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Double centralizer theorem In the branch of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (m ...
*
Idealizer In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
* Multipliers and centralizers (Banach spaces) *
Stabilizer subgroup In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

# References

* * * {{DEFAULTSORT:Centralizer And Normalizer Abstract algebra Group theory Ring theory Lie algebras