# Centralizer and normalizer

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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 of *G* that commute with *S* "as a whole". 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.

## Definitions

- Groups and semigroups

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

Sometimes if there is no ambiguity about the group in question, the *G* is suppressed from the notation entirely. When *S*={*a*} is a singleton set, then C_{G}({*a*}) can be abbreviated to C_{G}(*a*). Another less common notation for the centralizer is Z(*a*), which parallels the notation for the center of a group. 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*, given by Z(*g*).

The **normalizer** of *S* in the group (or semigroup) *G* is defined to be

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*, however if *g* is in the normalizer, *gs* = *tg* for some *t* in *S*, potentially different from *s*. The same conventions mentioned previously about suppressing *G* and suppressing braces from singleton sets also apply to the normalizer notation. The normalizer should not be confused with the normal closure.

- Rings, algebras, Lie rings and Lie algebras

If *R* is a ring or an algebra, and *S* is a subset of the ring, 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 beTemplate:Sfn

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 byTemplate:Sfn

While this is the standard usage of the term "normalizer" in Lie algebra, it should be noted that 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.Template:Sfn

## Properties

### Semigroups

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Let *S*′ be the centralizer, i.e. Then:

*S*′ forms a subsemigroup.- - A commutant is its own bicommutant.

### Groups

Source: Template:Sfn

- 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 will occur if*st*=*ts*for every*s*and*t*in*S*. Naturally then if*H*is an abelian subgroup of*G*,**C**_{G}(H) contains*H*.- If
*S*is a subsemigroup of*G*, then**N**_{G}(S) contains*S*. - If
*H*is a subgroup of*G*, then the largest subgroup in which*H*is normal is the subgroup**N**_{G}(H). - 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 automorphism group 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*is*T*(**C**_{G}(*S*)).

### Rings and algebras

Source: Template:Sfn

- Centralizers in rings and algebras are subrings and subalgebras, respectively, and centralizers in Lie rings and 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 also

- Commutator
- Stabilizer subgroup
- Multipliers and centralizers (Banach spaces)
- Double centralizer theorem
- Idealizer

## Notes

## References

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