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 are 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.

Definitions

Groups and semigroups

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

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

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 CG({a}) can be abbreviated to CG(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

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

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 ${\displaystyle {\mathfrak {L}}}$ is a Lie algebra (or Lie ring) with Lie product [x,y], then the centralizer of a subset S 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. If R is an associative ring, then R can be given the bracket product [x,y] = xyyx. Of course then xy = yx if and only if [x,y] = 0. If we denote the set R with the bracket product as LR, then clearly the ring centralizer of S in R is equal to the Lie ring centralizer of S in LR.

The normalizer of a subset S 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, it should be noted that this construction is actually the idealizer of the set S in ${\displaystyle {\mathfrak {L}}}$. If S 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 which S is a Lie ideal.[5]

Properties

Semigroups

Let S′ be the centralizer, i.e. ${\displaystyle S'=\{x\in A:sx=xs\ {\mbox{for}}\ {\mbox{every}}\ s\in S\}.}$ Then:

• S′ forms a subsemigroup.
• ${\displaystyle S'=S'''=S'''''}$ – A commutant is its own bicommutant.

Groups

Source: [6]

• The centralizer and normalizer of S are both subgroups of G.
• Clearly, CG(S) ⊆ NG(S). In fact, CG(S) is always a normal subgroup of NG(S).
• CG(CG(S)) contains S, but CG(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, CG(H) contains H.
• If S is a subsemigroup of G, then NG(S) contains S.
• If H is a subgroup of G, then the largest subgroup in which H is normal is the subgroup NG(H).
• A subgroup H of a group G is called a self-normalizing subgroup of G if NG(H) = H.
• The center of G is exactly CG(G) and G is an abelian group if and only if CG(G)=Z(G) = G.
• For singleton sets, CG(a)=NG(a).
• By symmetry, if S and T are two subsets of G, T ⊆ CG(S) if and only if S ⊆ CG(T).
• For a subgroup H of group G, the N/C theorem states that the factor group NG(H)/CG(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H. Since NG(G) = G and CG(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) = Tx(g) = xgx −1, then we can describe NG(S) and CG(S) in terms of the group action of Inn(G) on G: the stabilizer of S in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing S is T(CG(S)).
• A subgroup H of a group G is said to be C-closed or self-bicommutant if H = CG(S) for some subset S ⊆ G. If so, then in fact, H = CG(CG(H)).

Rings and algebras

Source: [4]

• 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.
• CR(CR(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 NA(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 ⊆ NA(S).

Notes

1. ^ 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.
2. ^ 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.
3. ^ Jacobson (2009), p. 41
4. ^ a b c Jacobson 1979, p.28.
5. ^ Jacobson 1979, p.57.
6. ^ Isaacs 2009, Chapters 1−3.