Centralizer and normalizer

<h2 id="definitions">Definitions</h2>
<h3>Group and semigroup</h3>
The centralizer of a subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of group (or semigroup) G is defined as<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

C
          
          
            G
          
        
        (
        S
        )
        =
        
          {
          
            g
            ∈
            G
            ∣
            g
            s
            =
            s
            g
            
               for all 
            
            s
            ∈
            S
          
          }
        
        =
        
          {
          
            g
            ∈
            G
            ∣
            g
            s
            
              g
              
                −
                1
              
            
            =
            s
            
               for all 
            
            s
            ∈
            S
          
          }
        
        ,
      
    
    {\displaystyle \mathrm {C} _{G}(S)=\left\{g\in G\mid gs=sg{\text{ for all }}s\in S\right\}=\left\{g\in G\mid gsg^{-1}=s{\text{ for all }}s\in S\right\},}

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
 =
 {
 a
 }
 
 
 {\displaystyle S=\{a\}}
 
 is a <a href="/facts/Singleton_(mathematics)/LtNVfujT">singleton</a> set, we write CG(a) instead of CG({a}). Another less common notation for the centralizer is Z(a), which parallels the notation for the <a href="/facts/Center_(group_theory)/gqAg6B8I">center</a>. 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

N
          
          
            G
          
        
        (
        S
        )
        =
        
          {
          
            g
            ∈
            G
            ∣
            g
            S
            =
            S
            g
          
          }
        
        =
        
          {
          
            g
            ∈
            G
            ∣
            g
            S
            
              g
              
                −
                1
              
            
            =
            S
          
          }
        
        ,
      
    
    {\displaystyle \mathrm {N} _{G}(S)=\left\{g\in G\mid gS=Sg\right\}=\left\{g\in G\mid gSg^{-1}=S\right\},}

where again only the first definition applies to semigroups. If the set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a subgroup of 
 
 
 
 G
 
 
 {\displaystyle G}
 
, then the normalizer 
 
 
 
 
 N
 
 G
 
 
 (
 S
 )
 
 
 {\displaystyle N_{G}(S)}
 
 is the largest subgroup 
 
 
 
 
 G
 ′
 
 ⊆
 G
 
 
 {\displaystyle G'\subseteq G}
 
 where 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of 
 
 
 
 
 G
 ′
 
 
 
 {\displaystyle G'}
 
. The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 and s is in 
 
 
 
 S
 
 
 {\displaystyle S}
 
, then it must be that gs = sg, but if g is in the normalizer, then gs = tg for some t in 
 
 
 
 S
 
 
 {\displaystyle S}
 
, with t possibly different from s. That is, elements of the centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 must commute pointwise with 
 
 
 
 S
 
 
 {\displaystyle 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 <a href="/facts/Conjugate_closure/k2lJahCW">normal closure</a>.
Clearly 
 
 
 
 
 C
 
 G
 
 
 (
 S
 )
 ⊆
 
 N
 
 G
 
 
 (
 S
 )
 
 
 {\displaystyle C_{G}(S)\subseteq N_{G}(S)}
 
 and both are subgroups of 
 
 
 
 G
 
 
 {\displaystyle G}
 
.

<h3>Ring, algebra over a field, Lie ring, and Lie algebra</h3>
If R is a ring or an <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra over a field</a>, and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a subset of R, then the centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is exactly as defined for groups, with R in the place of G.
If 
 
 
 
 
 
 L
 
 
 
 
 {\displaystyle {\mathfrak {L}}}
 
 is a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> (or <a href="/facts/Lie_ring/n2gAUkNq">Lie ring</a>) with Lie product [x, y], then the centralizer of a subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of 
 
 
 
 
 
 L
 
 
 
 
 {\displaystyle {\mathfrak {L}}}
 
 is defined to be<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

C
          
          
            
              L
            
          
        
        (
        S
        )
        =
        {
        x
        ∈
        
          
            L
          
        
        ∣
        [
        x
        ,
        s
        ]
        =
        0
        
           for all 
        
        s
        ∈
        S
        }
        .
      
    
    {\displaystyle \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 <a href="/facts/Commutator/eYxzFmFY">bracket product</a> [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
 
 
 {\displaystyle S}
 
 in R is equal to the Lie ring centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 in LR.
The normalizer of a subset 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of a Lie algebra (or Lie ring) 
 
 
 
 
 
 L
 
 
 
 
 {\displaystyle {\mathfrak {L}}}
 
 is given by<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

N
          
          
            
              L
            
          
        
        (
        S
        )
        =
        {
        x
        ∈
        
          
            L
          
        
        ∣
        [
        x
        ,
        s
        ]
        ∈
        S
        
           for all 
        
        s
        ∈
        S
        }
        .
      
    
    {\displaystyle \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, this construction is actually the <a href="/facts/Idealizer/A6xVTD4m">idealizer</a> of the set 
 
 
 
 S
 
 
 {\displaystyle S}
 
 in 
 
 
 
 
 
 L
 
 
 
 
 {\displaystyle {\mathfrak {L}}}
 
. If 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is an additive subgroup of 
 
 
 
 
 
 L
 
 
 
 
 {\displaystyle {\mathfrak {L}}}
 
, then 
 
 
 
 
 
 N
 
 
 
 L
 
 
 
 (
 S
 )
 
 
 {\displaystyle \mathrm {N} _{\mathfrak {L}}(S)}
 
 is the largest Lie subring (or Lie subalgebra, as the case may be) in which 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a Lie <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="example">Example</h2>
Consider the group

G
 =
 
 S
 
 3
 
 
 =
 {
 [
 1
 ,
 2
 ,
 3
 ]
 ,
 [
 1
 ,
 3
 ,
 2
 ]
 ,
 [
 2
 ,
 1
 ,
 3
 ]
 ,
 [
 2
 ,
 3
 ,
 1
 ]
 ,
 [
 3
 ,
 1
 ,
 2
 ]
 ,
 [
 3
 ,
 2
 ,
 1
 ]
 }
 
 
 {\displaystyle G=S_{3}=\{[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]\}}
 
 (the symmetric group of permutations of 3 elements).
Take a subset 
 
 
 
 H
 
 
 {\displaystyle H}
 
 of the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
:

H
        =
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        1
        ,
        3
        ,
        2
        ]
        }
        .
      
    
    {\displaystyle H=\{[1,2,3],[1,3,2]\}.}

Note that 
 
 
 
 [
 1
 ,
 2
 ,
 3
 ]
 
 
 {\displaystyle [1,2,3]}
 
 is the identity permutation in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and retains the order of each element and 
 
 
 
 [
 1
 ,
 3
 ,
 2
 ]
 
 
 {\displaystyle [1,3,2]}
 
 is the permutation that fixes the first element and swaps the second and third element.
The normalizer of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 with respect to the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 are all elements of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 that yield the set 
 
 
 
 H
 
 
 {\displaystyle H}
 
 (potentially permuted) when the element conjugates 
 
 
 
 H
 
 
 {\displaystyle H}
 
.
Working out the example for each element of 
 
 
 
 G
 
 
 {\displaystyle G}
 
:

[
        1
        ,
        2
        ,
        3
        ]
      
    
    {\displaystyle [1,2,3]}
  
 when applied to 
  
    
      
        H
      
    
    {\displaystyle H}
  
: 
  
    
      
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        1
        ,
        3
        ,
        2
        ]
        }
        =
        H
      
    
    {\displaystyle \{[1,2,3],[1,3,2]\}=H}
  
; therefore 
  
    
      
        [
        1
        ,
        2
        ,
        3
        ]
      
    
    {\displaystyle [1,2,3]}
  
 is in the normalizer 
  
    
      
        
          N
          
            G
          
        
        (
        H
        )
      
    
    {\displaystyle N_{G}(H)}
  
.

[
        1
        ,
        3
        ,
        2
        ]
      
    
    {\displaystyle [1,3,2]}
  
 when applied to 
  
    
      
        H
      
    
    {\displaystyle H}
  
: 
  
    
      
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        1
        ,
        3
        ,
        2
        ]
        }
        =
        H
      
    
    {\displaystyle \{[1,2,3],[1,3,2]\}=H}
  
; therefore 
  
    
      
        [
        1
        ,
        3
        ,
        2
        ]
      
    
    {\displaystyle [1,3,2]}
  
 is in the normalizer 
  
    
      
        
          N
          
            G
          
        
        (
        H
        )
      
    
    {\displaystyle N_{G}(H)}
  
.

[
        2
        ,
        1
        ,
        3
        ]
      
    
    {\displaystyle [2,1,3]}
  
 when applied to 
  
    
      
        H
      
    
    {\displaystyle H}
  
: 
  
    
      
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        3
        ,
        2
        ,
        1
        ]
        }
        ≠
        H
      
    
    {\displaystyle \{[1,2,3],[3,2,1]\}\neq H}
  
; therefore 
  
    
      
        [
        2
        ,
        1
        ,
        3
        ]
      
    
    {\displaystyle [2,1,3]}
  
 is not in the normalizer 
  
    
      
        
          N
          
            G
          
        
        (
        H
        )
      
    
    {\displaystyle N_{G}(H)}
  
.

[
        2
        ,
        3
        ,
        1
        ]
      
    
    {\displaystyle [2,3,1]}
  
 when applied to 
  
    
      
        H
      
    
    {\displaystyle H}
  
: 
  
    
      
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        2
        ,
        1
        ,
        3
        ]
        }
        ≠
        H
      
    
    {\displaystyle \{[1,2,3],[2,1,3]\}\neq H}
  
; therefore 
  
    
      
        [
        2
        ,
        3
        ,
        1
        ]
      
    
    {\displaystyle [2,3,1]}
  
 is not in the normalizer 
  
    
      
        
          N
          
            G
          
        
        (
        H
        )
      
    
    {\displaystyle N_{G}(H)}
  
.

[
        3
        ,
        1
        ,
        2
        ]
      
    
    {\displaystyle [3,1,2]}
  
 when applied to 
  
    
      
        H
      
    
    {\displaystyle H}
  
: 
  
    
      
        {
        [
        1
        ,
        2
        ,
        3
        ]
        ,
        [
        3
        ,
        2
        ,
        1
        ]
        }
        ≠
        H
      
    
    {\displaystyle \{[1,2,3],[3,2,1]\}\neq H}
  
; therefore 
  
    
      
        [
        3
        ,
        1
        ,
        2
        ]
      
    
    {\displaystyle [3,1,2]}
  
 is not in the normalizer 
  
    
      
        
          N
          
            G
          
        
        (
        H
        )
      
    
    {\displaystyle N_{G}(H)}
  
.

[
 3
 ,
 2
 ,
 1
 ]
 
 
 {\displaystyle [3,2,1]}
 
 when applied to 
 
 
 
 H
 
 
 {\displaystyle H}
 
: 
 
 
 
 {
 [
 1
 ,
 2
 ,
 3
 ]
 ,
 [
 2
 ,
 1
 ,
 3
 ]
 }
 ≠
 H
 
 
 {\displaystyle \{[1,2,3],[2,1,3]\}\neq H}
 
; therefore 
 
 
 
 [
 3
 ,
 2
 ,
 1
 ]
 
 
 {\displaystyle [3,2,1]}
 
 is not in the normalizer 
 
 
 
 
 N
 
 G
 
 
 (
 H
 )
 
 
 {\displaystyle N_{G}(H)}
 
.
Therefore, the normalizer 
 
 
 
 
 N
 
 G
 
 
 (
 H
 )
 
 
 {\displaystyle N_{G}(H)}
 
 of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 in 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is 
 
 
 
 {
 [
 1
 ,
 2
 ,
 3
 ]
 ,
 [
 1
 ,
 3
 ,
 2
 ]
 }
 
 
 {\displaystyle \{[1,2,3],[1,3,2]\}}
 
 since both these group elements preserve the set 
 
 
 
 H
 
 
 {\displaystyle H}
 
 under conjugation.
The centralizer of the group 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is the set of elements that leave each element of 
 
 
 
 H
 
 
 {\displaystyle H}
 
 unchanged by conjugation; that is, the set of elements that commutes with every element in 
 
 
 
 H
 
 
 {\displaystyle H}
 
. 
It's clear in this example that the only such element in S3 is 
 
 
 
 H
 
 
 {\displaystyle H}
 
 itself ([1, 2, 3], [1, 3, 2]).

<h2 id="properties">Properties</h2>
<h3>Semigroups</h3>
Let 
 
 
 
 
 S
 ′
 
 
 
 {\displaystyle S'}
 
 denote the centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 in the semigroup 
 
 
 
 A
 
 
 {\displaystyle A}
 
; i.e. 
 
 
 
 
 S
 ′
 
 =
 {
 x
 ∈
 A
 ∣
 s
 x
 =
 x
 s
 
  for every 
 
 s
 ∈
 S
 }
 .
 
 
 {\displaystyle S'=\{x\in A\mid sx=xs{\text{ for every }}s\in S\}.}
 
 Then 
 
 
 
 
 S
 ′
 
 
 
 {\displaystyle S'}
 
 forms a <a href="/facts/Subsemigroup/rgSdk2kt">subsemigroup</a> and 
 
 
 
 
 S
 ′
 
 =
 
 S
 ‴
 
 =
 
 S
 ′′′′′
 
 
 
 {\displaystyle S'=S'''=S'''''}
 
; i.e. a commutant is its own <a href="/facts/Bicommutant/kNCH5yHy">bicommutant</a>.

<h3>Groups</h3>
Source:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<ul><li>The centralizer and normalizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 are both subgroups of G.</li>
<li>Clearly, CG(S) ⊆ NG(S). In fact, CG(S) is always a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of NG(S), being the kernel of the <a href="/facts/Group_homomorphism/sDguaNYs">homomorphism</a> NG(S) → Bij(S) and the group NG(S)/CG(S) <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acts</a> by conjugation as a <a href="/facts/Symmetric_group/RUksK5l5">group of bijections</a> on S. E.g. the <a href="/facts/Weyl_group/N9QG4Wyn">Weyl group</a> of a compact <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> G with a torus T is defined as W(G,T) = NG(T)/CG(T), and especially if the torus is maximal (i.e. CG(T) = T) it is a central tool in the theory of Lie groups.</li>
<li>CG(CG(S)) contains 
 
 
 
 S
 
 
 {\displaystyle S}
 
, but CG(S) need not contain 
 
 
 
 S
 
 
 {\displaystyle S}
 
. Containment occurs exactly when 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is abelian.</li>
<li>If H is a subgroup of G, then NG(H) contains H.</li>
<li>If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup NG(H).</li>
<li>If 
 
 
 
 S
 
 
 {\displaystyle 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
 
 
 {\displaystyle S}
 
 is the subgroup CG(S).</li>
<li>A subgroup H of a group G is called a self-normalizing subgroup of G if NG(H) = H.</li>
<li>The center of G is exactly CG(G) and G is an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> if and only if CG(G) = Z(G) = G.</li>
<li>For singleton sets, CG(a) = NG(a).</li>
<li>By symmetry, if 
 
 
 
 S
 
 
 {\displaystyle S}
 
 and T are two subsets of G, T ⊆ CG(S) if and only if S ⊆ CG(T).</li>
<li>For a subgroup H of group G, the N/C theorem states that the <a href="/facts/Factor_group/HlwpHtjb">factor group</a> NG(H)/CG(H) is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to a subgroup of Aut(H), the group of <a href="/facts/Automorphism/WnyxNFkQ">automorphisms</a> 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 <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphisms</a> of G.</li>
<li>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
 
 
 {\displaystyle S}
 
 in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing 
 
 
 
 S
 
 
 {\displaystyle S}
 
 pointwise is T(CG(S)).</li>
<li>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)).</li></ul>
<h3>Rings and algebras over a field</h3>
Source:<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<ul><li>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.</li>
<li>The normalizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 in a Lie ring contains the centralizer of 
 
 
 
 S
 
 
 {\displaystyle S}
 
.</li>
<li>CR(CR(S)) contains 
 
 
 
 S
 
 
 {\displaystyle S}
 
 but is not necessarily equal. The <a href="/facts/Double_centralizer_theorem/q5Lo1mgP">double centralizer theorem</a> deals with situations where equality occurs.</li>
<li>If 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is an additive subgroup of a Lie ring A, then NA(S) is the largest Lie subring of A in which 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a Lie ideal.</li>
<li>If 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is a Lie subring of a Lie ring A, then S ⊆ NA(S).</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Commutator/eYxzFmFY">Commutator</a></li>
<li><a href="/facts/Multipliers_and_centralizers_(Banach_spaces)/lWlludmB">Multipliers and centralizers (Banach spaces)</a></li>
<li><a href="/facts/Stabilizer_subgroup/Vh9VtUUw">Stabilizer subgroup</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Isaacs, I. Martin (2009), <a href="https://books.google.com/books?id=5tKq0kbHuc4C&q=centralizer+OR+normalizer">Algebra: a graduate course</a>, <a href="/facts/Graduate_Studies_in_Mathematics/APstozeU">Graduate Studies in Mathematics</a>, vol. 100 (reprint of the 1994 original ed.), Providence, RI: <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fgsm%2F100">10.1090/gsm/100</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-4799-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2472787">2472787</a></li>
<li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (2009), <a href="https://books.google.com/books?id=qAg_AwAAQBAJ&q=centralizer+OR+normalizer">Basic Algebra</a>, vol. 1 (2 ed.), <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-47189-1</li>
<li>Jacobson, Nathan (1979), <a href="https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer">Lie Algebras</a> (republication of the 1962 original ed.), <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-63832-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0559927">0559927</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="978-0-19-979373-0" target="_blank">978-0-19-979373-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="978-3-03719-032-6" target="_blank">978-3-03719-032-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Jacobson (2009), p. 41 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Jacobson 1979, p. 28. - Jacobson, Nathan (1979), Lie Algebras (republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927 <a href="https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer" target="_blank">https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Jacobson 1979, p. 28. - Jacobson, Nathan (1979), Lie Algebras (republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927 <a href="https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer" target="_blank">https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Jacobson 1979, p. 57. - Jacobson, Nathan (1979), Lie Algebras (republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927 <a href="https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer" target="_blank">https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Isaacs 2009, Chapters 1−3. - Isaacs, I. Martin (2009), Algebra: a graduate course, Graduate Studies in Mathematics, vol. 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 <a href="https://books.google.com/books?id=5tKq0kbHuc4C&q=centralizer+OR+normalizer" target="_blank">https://books.google.com/books?id=5tKq0kbHuc4C&q=centralizer+OR+normalizer</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Jacobson 1979, p. 28. - Jacobson, Nathan (1979), Lie Algebras (republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927 <a href="https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer" target="_blank">https://books.google.com/books?id=hPE1Mmm7SFMC&q=centralizer+OR+normalizer</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Centralizer and normalizer open-in-new

Centralizer and normalizer