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 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.


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 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

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, but if g is in the normalizer, then gs = tg for some t in S, with t 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] = 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) 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]



Let denote the centralizer of in the semigroup , i.e. Then forms a subsemigroup and , i.e. a commutant is its own bicommutant.



  • 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 occurs exactly when S is abelian.
  • If H is a subgroup of G, then NG(H) contains H.
  • If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup NG(H).
  • If S is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S is the subgroup CG(S).
  • 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 pointwise 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 over a field


  • 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.
  • 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).

See also


  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. Jacobson 1979, p.28.
  5. Jacobson 1979, p.57.
  6. Isaacs 2009, Chapters 1−3.


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