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.
Contents
Definitions
Group and semigroup
The centralizer of a subset S of group (or semigroup) G is defined to be^{[3]}
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
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. That is, elements of the centralizer of S must commute pointwise with S, but elements of the normalizer of S need only commute with S as a set. 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 is a Lie algebra (or Lie ring) with Lie product [x,y], then the centralizer of a subset S of is defined to be^{[4]}
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) is given by^{[4]}
While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set S in . If S is an additive subgroup of , then is the largest Lie subring (or Lie subalgebra, as the case may be) in which S is a Lie ideal.^{[5]}
Properties
Semigroups
Let denote the centralizer of in the semigroup , i.e. Then:
- forms a subsemigroup.
- —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).
See also
- Commutator
- Double centralizer theorem
- Idealizer
- Multipliers and centralizers (Banach spaces)
- Stabilizer subgroup
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
- ^ ^{a} ^{b} ^{c} 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