Centralizer and normalizer

"Normalizer" redirects here. For the process of increasing audio amplitude, see Audio normalization.

"Centralizer" redirects here. For centralizers of Banach spaces, see Multipliers and centralizers (Banach spaces).

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 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 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, 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).

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, 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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