Inmathematics, especiallygroup theory, thecentralizer (also calledcommutant^[1][2]) of asubsetS in agroupG is the set${\displaystyle {\mathrm{C}}_G(S)}$ of elements ofG thatcommute with every element ofS, or equivalently, the set of elements${\displaystyle g\in G}$ such thatconjugation by${\displaystyle g}$ leaves each element ofS fixed. Thenormalizer ofS inG is theset of elements${\displaystyle {\mathrm{N}}_G(S)}$ ofG that satisfy the weaker condition of leaving the set${\displaystyle S\subseteq G}$ fixed under conjugation. The centralizer and normalizer ofS aresubgroups ofG. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S.

Suitably formulated, the definitions also apply tosemigroups.

Inring theory, thecentralizer of a subset of aring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ringR is asubring ofR. This article also deals with centralizers and normalizers in aLie algebra.

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

Thecentralizer of a subsetS of group (or semigroup)G is defined as^[3]

${\displaystyle {\mathrm{C}}_G(S)=\left\{g\in G\mid gs=sg\text{ for all }s\in S\right\}=\left\{g\in G\mid gs{g}^{-1}=s\text{ for all }s\in S\right\},}$

where only the first definition applies to semigroups.If there is no ambiguity about the group in question, theG can be suppressed from the notation. WhenS = {a} is asingleton 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 thecenter. With this latter notation, one must be careful to avoid confusion between thecenter of a groupG, Z(G), and thecentralizer of anelementg inG, Z(g).

Thenormalizer ofS in the group (or semigroup)G is defined as

${\displaystyle {\mathrm{N}}_G(S)=\left\{g\in G\mid gS=Sg\right\}=\left\{g\in G\mid gS{g}^{-1}=S\right\},}$

where again only the first definition applies to semigroups. If the set${\displaystyle S}$ is a subgroup of${\displaystyle G}$, then the normalizer${\displaystyle N_G(S)}$ is the largest subgroup${\displaystyle G^{\prime }\subseteq G}$ where${\displaystyle S}$ is anormal subgroup of${\displaystyle G^{\prime }}$. The definitions ofcentralizer andnormalizer are similar but not identical. Ifg is in the centralizer ofS ands is inS, then it must be thatgs =sg, but ifg is in the normalizer, thengs =tg for somet inS, witht possibly different froms. That is, elements of the centralizer ofS must commute pointwise withS, but elements of the normalizer ofS need only commute withS as a set. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with thenormal closure.

Clearly${\displaystyle C_G(S)\subseteq N_G(S)}$ and both are subgroups of${\displaystyle G}$.

IfR is a ring or analgebra over a field, andS is a subset ofR, then the centralizer ofS is exactly as defined for groups, withR in the place ofG.

If${\displaystyle \mathfrak{L}}$ is aLie algebra (orLie ring) with Lie product [x,y], then the centralizer of a subsetS of${\displaystyle \mathfrak{L}}$ is defined to be^[4]

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

The definition of centralizers for Lie rings is linked to the definition for rings in the following way. IfR is an associative ring, thenR can be given thebracket product[x,y] =xy −yx. Of course thenxy =yx if and only if[x,y] = 0. If we denote the setR with the bracket product as L_R, then clearly thering centralizer ofS inR is equal to theLie ring centralizer ofS in L_R.

The normalizer of a subsetS of a Lie algebra (or Lie ring)${\displaystyle \mathfrak{L}}$ is given by^[4]

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

While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually theidealizer of the setS in${\displaystyle \mathfrak{L}}$. IfS is an additive subgroup of${\displaystyle \mathfrak{L}}$, then${\displaystyle {\mathrm{N}}_{\mathfrak{L}}(S)}$ is the largest Lie subring (or Lie subalgebra, as the case may be) in whichS is a Lieideal.^[5]

Consider the group

${\displaystyle G=S_3=\{[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]\}}$ (the symmetric group of permutations of 3 elements).

Take a subset H of the group G:

${\displaystyle H=\{[1,2,3],[1,3,2]\}.}$

Note that [1, 2, 3] is the identity permutation in G and retains the order of each element and [1, 3, 2] is the permutation that fixes the first element and swaps the second and third element.

The normalizer of H with respect to the group G are all elements of G that yield the set H (potentially permuted) when the element conjugates H.Working out the example for each element of G:

${\displaystyle [1,2,3]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[1,3,2]\}=H}$; therefore [1, 2, 3] is in the Normalizer(H) with respect to G.

${\displaystyle [1,3,2]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[1,3,2]\}=H}$; therefore [1, 3, 2] is in the Normalizer(H) with respect to G.

${\displaystyle [2,1,3]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[3,2,1]\}\ne H}$; therefore [2, 1, 3] is not in the Normalizer(H) with respect to G.

${\displaystyle [2,3,1]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[2,1,3]\}\ne H}$; therefore [2, 3, 1] is not in the Normalizer(H) with respect to G.

${\displaystyle [3,1,2]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[3,2,1]\}\ne H}$; therefore [3, 1, 2] is not in the Normalizer(H) with respect to G.

${\displaystyle [3,2,1]}$ when applied to${\displaystyle H}$:${\displaystyle \{[1,2,3],[2,1,3]\}\ne H}$; therefore [3, 2, 1] is not in the Normalizer(H) with respect to G.

Therefore, the Normalizer(H) with respect to G is${\displaystyle \{[1,2,3],[1,3,2]\}}$ since both these group elements preserve the set H under conjugation.

The centralizer of the group G is the set of elements that leave each element of H unchanged by conjugation; that is, the set of elements that commutes with every element in H. It's clear in this example that the only such element in S₃ is H itself ([1, 2, 3], [1, 3, 2]).

Let${\displaystyle S^{\prime }}$ denote the centralizer of${\displaystyle S}$ in the semigroup${\displaystyle A}$; i.e.${\displaystyle S^{\prime }=\{x\in A\mid sx=xs\text{ for every }s\in S\}.}$ Then${\displaystyle S^{\prime }}$ forms asubsemigroup and${\displaystyle S^{\prime }=S^{‴}=S^{\prime \prime \prime \prime \prime }}$; i.e. a commutant is its ownbicommutant.

Source:^[6]

• The centralizer and normalizer ofS are both subgroups ofG.
• Clearly,C_G(S) ⊆ N_G(S). In fact, C_G(S) is always anormal subgroup of N_G(S), being the kernel of thehomomorphismN_G(S) → Bij(S) and the group N_G(S)/C_G(S)acts by conjugation as agroup of bijections onS. E.g. theWeyl group of a compactLie groupG with a torusT is defined asW(G,T) = N_G(T)/C_G(T), and especially if the torus is maximal (i.e.C_G(T) =T) it is a central tool in the theory of Lie groups.
• C_G(C_G(S)) containsS, but C_G(S) need not containS. Containment occurs exactly whenS is abelian.
• IfH is a subgroup ofG, then N_G(H) containsH.
• IfH is a subgroup ofG, then the largest subgroup ofG in whichH is normal is the subgroup N_G(H).
• IfS is a subset ofG such that all elements ofS commute with each other, then the largest subgroup ofG whose center containsS is the subgroup C_G(S).
• A subgroupH of a groupG is called aself-normalizing subgroup ofG ifN_G(H) =H.
• The center ofG is exactly C_G(G) andG is anabelian group if and only ifC_G(G) = Z(G) =G.
• For singleton sets,C_G(a) = N_G(a).
• By symmetry, ifS andT are two subsets ofG,T ⊆ C_G(S) if and only ifS ⊆ C_G(T).
• For a subgroupH of groupG, theN/C theorem states that thefactor group N_G(H)/C_G(H) isisomorphic to a subgroup of Aut(H), the group ofautomorphisms ofH. SinceN_G(G) =G andC_G(G) = Z(G), the N/C theorem also implies thatG/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of allinner automorphisms ofG.
• If we define a group homomorphismT :G → Inn(G) byT(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) onG: the stabilizer ofS in Inn(G) isT(N_G(S)), and the subgroup of Inn(G) fixingS pointwise isT(C_G(S)).
• A subgroupH of a groupG is said to beC-closed orself-bicommutant ifH = C_G(S) for some subsetS ⊆G. If so, then in fact,H = C_G(C_G(H)).

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 ofS in a Lie ring contains the centralizer ofS.
• C_R(C_R(S)) containsS but is not necessarily equal. Thedouble centralizer theorem deals with situations where equality occurs.
• IfS is an additive subgroup of a Lie ringA, then N_A(S) is the largest Lie subring ofA in whichS is a Lie ideal.
• IfS is a Lie subring of a Lie ringA, thenS ⊆ N_A(S).

• Commutator
• Multipliers and centralizers (Banach spaces)
• Stabilizer subgroup

• Isaacs, I. Martin (2009),Algebra: a graduate course,Graduate Studies in Mathematics, vol. 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, vol. 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

• Abstract algebra
• Group theory
• Ring theory
• Lie algebras

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