# Centralizer and normalizer

In mathematics, especially group theory, the **centralizer** (also called **commutant**^{[1]}^{[2]}) of a subset *S* of a group *G* is the set of elements of *G* that commute with each element of *S*, and the **normalizer** of *S* is the set of elements that satisfy a weaker condition. The centralizer and normalizer of *S* are subgroups of *G*, and can provide insight into the structure of *G*.

The definitions also apply to monoids and semigroups.

In ring theory, 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 of *R*. This article also deals with centralizers and normalizers in Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

## DefinitionsEdit

### Group and semigroupEdit

The **centralizer** of a subset *S* of group (or semigroup) *G* is defined as^{[3]}

If there is no ambiguity about the group in question, the *G* can be suppressed from the notation. When *S* = {*a*} is a singleton set, we write C_{G}(*a*) instead of C_{G}({*a*}). 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

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 *gs* = *sg*, but if *g* is in the normalizer, then *gs* = *tg* 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.

### Ring, algebra over a field, Lie ring, and Lie algebraEdit

If *R* is a ring or an algebra over a field, 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 (or Lie ring) with Lie product [*x*,*y*], then the centralizer of a subset *S* of is defined to be^{[4]}

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 [*x*,*y*] = *xy* − *yx*. Of course then *xy* = *yx* if and only if [*x*,*y*] = 0. 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^{[4]}

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.^{[5]}

## PropertiesEdit

### SemigroupsEdit

Let denote the centralizer of in the semigroup , i.e. Then forms a subsemigroup and , i.e. a commutant is its own bicommutant.

### GroupsEdit

Source:^{[6]}

- The centralizer and normalizer of
*S*are both subgroups of*G*. - Clearly,
**C**_{G}(*S*) ⊆**N**_{G}(*S*). In fact,**C**_{G}(*S*) is always a normal subgroup of**N**_{G}(*S*). **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**self-normalizing subgroup**of*G*if**N**_{G}(*H*) =*H*. - The center of
*G*is exactly**C**_{G}(G) and*G*is an abelian group 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**N**_{G}(*H*)/**C**_{G}(*H*) is isomorphic to a subgroup of Aut(*H*), the group of automorphisms 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 automorphisms of*G*. - If we define a group homomorphism
*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 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 fieldEdit

Source:^{[4]}

- 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*S*⊆**N**_{A}(*S*).

## See alsoEdit

## NotesEdit

**^**Kevin O'Meara; John Clark; Charles Vinsonhaler (2011).*Advanced Topics in Linear Algebra: Weaving Matrix Problems Through the Weyr Form*. Oxford University Press. p. 65. ISBN 978-0-19-979373-0.**^**Karl Heinrich Hofmann; Sidney A. Morris (2007).*The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups*. European Mathematical Society. p. 30. ISBN 978-3-03719-032-6.**^**Jacobson (2009), p. 41- ^
^{a}^{b}^{c}Jacobson 1979, p.28. **^**Jacobson 1979, p.57.**^**Isaacs 2009, Chapters 1−3.

## ReferencesEdit

- Isaacs, I. Martin (2009),
*Algebra: a graduate course*, Graduate Studies in Mathematics,**100**(reprint of the 1994 original ed.), Providence, RI: American Mathematical Society, doi:10.1090/gsm/100, ISBN 978-0-8218-4799-2, MR 2472787 - Jacobson, Nathan (2009),
*Basic Algebra*,**1**(2 ed.), Dover Publications, ISBN 978-0-486-47189-1 - Jacobson, Nathan (1979),
*Lie Algebras*(republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927