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 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

Group and semigroup

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

\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 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.

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

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 ${\mathfrak {L}}$ 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 be^[4]

{\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 by^[4]

{\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 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.^[5]

Properties

Semigroups

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

$S'$ forms a subsemigroup.
$S'=S'''=S'''''$ —i.e. a commutant is its own bicommutant.

Groups

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 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 H is a subgroup of G, then N_G(H) contains H.
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 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 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 field

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).

Notes

↑ 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
1 2 3 Jacobson 1979, p.28.
↑ Jacobson 1979, p.57.
↑ Isaacs 2009, Chapters 1−3.

References

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

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[O'MearaClark2011-1] 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.

[HofmannMorris2007-2] 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

[FOOTNOTEJacobson1979p.28-4] 1 2 3 Jacobson 1979, p.28.

[FOOTNOTEJacobson1979p.57-5] Jacobson 1979, p.57.

[FOOTNOTEIsaacs2009Chapters_1−3-6] Isaacs 2009, Chapters 1−3.