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# 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 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]

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

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 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

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

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 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 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) \( \mathfrak{L} \) 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, 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.[5]

Properties

Semigroups

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

S′ forms a subsemigroup.

S' = S''' = S''''' - 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
*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*)). - A subgroup
*H*of a group*G*is said to be**C-closed**if*H*=**C**_{G}(*S*) for some subset*S*⊆*G*. If so, then in fact,*H*=**C**_{G}(**C**_{G}(*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*. **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

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

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, pp. xii+516, ISBN 978-0-8218-4799-2, MR 2472787

Jacobson, Nathan (2009), Basic algebra 1 (2 ed.), Dover, ISBN 978-0-486-47189-1

Jacobson, Nathan (1979), Lie algebras (republication of the 1962 original ed.), New York: Dover Publications Inc., pp. ix+331, ISBN 0-486-63832-4, MR 559927

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