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In mathematics, especially group theory, the centralizer 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

$\mathrm {C} _{G}(S)=\{g\in G\mid sg=gs{\text{ for all }}s\in S\}$

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

$\mathrm {N} _{G}(S)=\{g\in G\mid gS=Sg\}$

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 ${\mathfrak {L}}$ is a Lie algebra (or Lie ring) with Lie product [x,y], then the centralizer of a subset S of ${\mathfrak {L}}$ is defined to beTemplate:Sfn

$\mathrm {C} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid [x,s]=0{\text{ for all }}s\in S\}$

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 [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) ${\mathfrak {L}}$ is given byTemplate:Sfn

$\mathrm {N} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid [x,s]\in S{\text{ for all }}s\in S\}$

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 ${\mathfrak {L}}$. If S is an additive subgroup of ${\mathfrak {L}}$, then $\mathrm {N} _{\mathfrak {L}}(S)$ is the largest Lie subring (or Lie subalgebra, as the case may be) in which S is a Lie ideal.Template:Sfn

Properties

GroupsTemplate: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 algebrasTemplate: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

• Commutant
• Stabilizer subgroup
• Multipliers and centralizers (Banach spaces)
• Double centralizer theorem
• Idealizer

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010