Centralizer and normalizer

 ( Abstract Algebra ) 

In mathematics, especially group theory, the centralizer (also called commutant

(2026). 9783037190326, European Mathematical Society. . ISBN 9783037190326

) of a subset S in a group G is the set $\backslash operatorname\{C\}\_G(S)$ of elements of G that commutativity with every element of S, or equivalently, the set of elements $g\backslash in\; G$ such that conjugation by $g$ leaves each element of S fixed. The normalizer of S in G is the set of elements $\backslash mathrm\{N\}\_G(S)$ of G that satisfy the weaker condition of leaving the set $S\; \backslash subseteq\; G$ fixed under conjugation. The centralizer and normalizer of S are of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets  S.

Suitably formulated, the definitions also apply to .

In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra.

The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer.

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Definitions

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Group and semigroup The centralizer of a subset $S$ of group (or semigroup) G is defined asJacobson (2009), p. 41

$\backslash mathrm\{C\}\_G(S)\; =\; \backslash left\backslash \{g\; \backslash in\; G\; \backslash mid\; gs\; =\; sg\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\backslash right\backslash \}\; =\; \backslash left\backslash \{g\; \backslash in\; G\; \backslash mid\; gsg^\{-1\}\; =\; s\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\backslash right\backslash \},$

where only the first definition applies to semigroups. If there is no ambiguity about the group in question, the G can be suppressed from the notation. When $S=\backslash \{a\backslash \}$ is a singleton set, we write C _G( a) instead of C _G({ a}). Another less common notation for the centralizer is Z( a), which parallels the notation for the center. 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, Z( g).

The normalizer of S in the group (or semigroup) G is defined as

$\backslash mathrm\{N\}\_G(S)\; =\; \backslash left\backslash \{\; g\; \backslash in\; G\; \backslash mid\; gS\; =\; Sg\; \backslash right\backslash \}\; =\; \backslash left\backslash \{g\; \backslash in\; G\; \backslash mid\; gSg^\{-1\}\; =\; S\backslash right\backslash \},$

where again only the first definition applies to semigroups. If the set $S$ is a subgroup of $G$, then the normalizer $N\_G(S)$ is the largest subgroup $G\text{'}\; \backslash subseteq\; G$ where $S$ is a normal subgroup of $G\text{'}$. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of $S$ and s is in $S$, then it must be that , but if g is in the normalizer, then for some t in $S$, with t possibly 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 notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure.

Clearly $C\_G(S)\; \backslash subseteq\; N\_G(S)$ and both are subgroups of $G$.

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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 $\backslash mathfrak\{L\}$ is a Lie algebra (or Lie ring) with Lie product x,, then the centralizer of a subset $S$ of $\backslash mathfrak\{L\}$ is defined to be

$\backslash mathrm\{C\}\_\{\backslash mathfrak\{L\}\}(S)\; =\; \backslash \{\; x\; \backslash in\; \backslash mathfrak\{L\}\; \backslash mid\; x,\; =\; 0\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\; \backslash \}.$

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 . Of course then if and only if . 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 Lie bracket can also be viewed as an operation of the set $\backslash mathfrak\{L\}$ on itself, because $*,*:\backslash mathfrak\{L\}\backslash times\backslash mathfrak\{L\}\backslash rightarrow\backslash mathfrak\{L\}$. The Lie bracket makes $(\backslash mathfrak\{L\},*,*)$ a group and its centralizer would then be all elements $\backslash \{\; x\; \backslash in\; \backslash mathfrak\{L\}\; \backslash mid\; x,\; =\; s,\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\; \backslash \}.$ However, since the Lie bracket is alternating, this condition is equivalent to $\backslash \{\; x\; \backslash in\; \backslash mathfrak\{L\}\; \backslash mid\; x,\; =\; 0\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\; \backslash \}.$ Thus, the centralizer is defined in the same way for Lie algebras as for groups.

The normalizer of a subset $S$ of a Lie algebra (or Lie ring) $\backslash mathfrak\{L\}$ is given by

$\backslash mathrm\{N\}\_\backslash mathfrak\{L\}(S)\; =\; \backslash \{\; x\; \backslash in\; \backslash mathfrak\{L\}\; \backslash mid\; x,\; \backslash in\; S\; \backslash text\{\; for\; all\; \}\; s\; \backslash in\; S\; \backslash \}.$

While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set $S$ in $\backslash mathfrak\{L\}$. If $S$ is an additive subgroup of $\backslash mathfrak\{L\}$, then $\backslash mathrm\{N\}\_\{\backslash mathfrak\{L\}\}(S)$ is the largest Lie subring (or Lie subalgebra, as the case may be) in which $S$ is a Lie ideal.

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Example Consider the group

$G\; =\; S\_3\; =\; \backslash \{1,,\; 1,,\; 2,,\; 2,,\; 3,,\; 3,\backslash \}$ (the symmetric group of permutations of 3 elements).

Take a subset $H$ of the group $G$:

$H\; =\; \backslash \{1,,\; 1,\backslash \}.$

Note that $1,$ is the identity permutation in $G$ and retains the order of each element and $1,$ is the permutation that fixes the first element and swaps the second and third element.

The normalizer of $H$ with respect to the group $G$ are all elements of $G$ that yield the set $H$ (potentially permuted) when the element conjugates $H$. Working out the example for each element of $G$:

$1,$ when applied to $H$: $\backslash \{1,,\; 1,\backslash \}\; =\; H$; therefore $1,$ is in the normalizer $N\_G(H)$.

$1,$ when applied to $H$: $\backslash \{1,,\; 1,\backslash \}\; =\; H$; therefore $1,$ is in the normalizer $N\_G(H)$.

$2,$ when applied to $H$: $\backslash \{1,,\; 3,\backslash \}\; \backslash neq\; H$; therefore $2,$ is not in the normalizer $N\_G(H)$.

$2,$ when applied to $H$: $\backslash \{1,,\; 2,\backslash \}\; \backslash neq\; H$; therefore $2,$ is not in the normalizer $N\_G(H)$.

$3,$ when applied to $H$: $\backslash \{1,,\; 3,\backslash \}\; \backslash neq\; H$; therefore $3,$ is not in the normalizer $N\_G(H)$.

$3,$ when applied to $H$: $\backslash \{1,,\; 2,\backslash \}\; \backslash neq\; H$; therefore $3,$ is not in the normalizer $N\_G(H)$.

Therefore, the normalizer $N\_G(H)$ of $H$ in $G$ is $\backslash \{1,,\; 1,\backslash \}$ since both these group elements preserve the set $H$ under conjugation.

The centralizer of the group $G$ is the set of elements that leave each element of $H$ unchanged by conjugation; that is, the set of elements that commutes with every element in $H$. It's clear in this example that the only such element in S₃ is $H$ itself (1,, 1,).

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Properties

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Semigroups Let $S\text{'}$ denote the centralizer of $S$ in the semigroup $A$; i.e. $S\text{'}\; =\; \backslash \{x\; \backslash in\; A\; \backslash mid\; sx\; =\; xs\; \backslash text\{\; for\; every\; \}\; s\; \backslash in\; S\backslash \}.$ Then $S\text{'}$ forms a subsemigroup and $S\text{'}\; =\; S$ = S''; i.e. a commutant is its own bicommutant.

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Groups Source:

• The centralizer and normalizer of $S$ are both subgroups of G.
• Clearly, . In fact, C _G( S) is always a normal subgroup of N _G( S), being the kernel of the homomorphism and the group N _G( S)/C _G( S) acts by conjugation as a Symmetric group on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as , and especially if the torus is maximal (i.e. it is a central tool in the theory of Lie groups.
• C _G(C _G( S)) contains $S$, but C _G( S) need not contain $S$. Containment occurs exactly when $S$ is abelian.
• If H is a subgroup of G, then N _G( H) contains H.
• If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup N _G( 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 C _G( S).
• A subgroup H of a group G is called a of G if .
• The center of G is exactly C _G(G) and G is an abelian group if and only if .
• For singleton sets, .
• By symmetry, if $S$ and T are two subsets of G, if and only if .
• 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 of H. Since and , 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 by , 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 $S$ pointwise is T(C _G( S)).
• A subgroup H of a group G is said to be C-closed or self-bicommutant if for some subset . If so, then in fact, .

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Rings and algebras over a field Source:

• 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 .

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See also

• Commutator
• Multipliers and centralizers (Banach spaces)
• Stabilizer subgroup

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Notes

Categories: Abstract Algebra, Group Theory, Ring Theory, Lie Algebras

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