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Difference between revisions of "Centralizer"

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If the magma <math>E</math> is [[associative]], then the centralizer of <math>X</math> is also the centralizer of the subset of <math>E</math> genererated by <math>X</math>, and the centralizer of <math>X</math> is furthermore an associative sub-magma of <math>E</math>.  If <math>E</math> is a [[group]], then the centralizer of <math>X</math> is a [[subgroup]], though not necessarily [[normal subgroup | normal]].  The centralizer of <math>E</math> is also called the ''[[center (algebra) | center]]'' of <math>E</math>.
 
If the magma <math>E</math> is [[associative]], then the centralizer of <math>X</math> is also the centralizer of the subset of <math>E</math> genererated by <math>X</math>, and the centralizer of <math>X</math> is furthermore an associative sub-magma of <math>E</math>.  If <math>E</math> is a [[group]], then the centralizer of <math>X</math> is a [[subgroup]], though not necessarily [[normal subgroup | normal]].  The centralizer of <math>E</math> is also called the ''[[center (algebra) | center]]'' of <math>E</math>.
  
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== Centralizers in Groups ==
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If <math>G</math> is a group, then an element <math>b</math> of <math>G</math> is said to ''centralize'' <math>A</math> if it commutes with every element of <math>A</math>; that is, if <math>bab^{-1} = a</math> for all <math>a \in A</math>.  A subset <math>B</math> of <math>G</math> is said to centralize <math>A</math> if all its elements centralize <math>A</math>.  The centralizer of <math>A</math>, denoted <math>C_G(A)</math>, or <math>C(A)</math> when there is no risk of confusion, is the set of elements that centralize <math>A</math>.  It is evidently a subgroup of <math>G</math>.
  
 
== See also ==
 
== See also ==
  
 
* [[Center (algebra)]]
 
* [[Center (algebra)]]
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* [[Normalizer]]
  
 
[[Category:Abstract algebra]]
 
[[Category:Abstract algebra]]

Latest revision as of 22:45, 14 May 2008

A centralizer is part of an algebraic structure.

Specifically, let $E$ be a magma, and let $X$ be a subset of $E$. The centralizer $X'$ of $X$ is the set of elements of $E$ which commute with every element of $X'$.

If $X \subseteq Y$ are subsets of a magma $E$, then $Y' \subseteq X'$. The bicentralizer $X''$ of $X$ is the centralizer of $X'$. Evidently, $X \subseteq X''$. The centralizer of the bicentralizer, $X'''$, is equal to $X'$, for $X' \subseteq X'''$, but $X \subseteq X''$, so $X''' \subseteq X'$.

If the magma $E$ is associative, then the centralizer of $X$ is also the centralizer of the subset of $E$ genererated by $X$, and the centralizer of $X$ is furthermore an associative sub-magma of $E$. If $E$ is a group, then the centralizer of $X$ is a subgroup, though not necessarily normal. The centralizer of $E$ is also called the center of $E$.

Centralizers in Groups

If $G$ is a group, then an element $b$ of $G$ is said to centralize $A$ if it commutes with every element of $A$; that is, if $bab^{-1} = a$ for all $a \in A$. A subset $B$ of $G$ is said to centralize $A$ if all its elements centralize $A$. The centralizer of $A$, denoted $C_G(A)$, or $C(A)$ when there is no risk of confusion, is the set of elements that centralize $A$. It is evidently a subgroup of $G$.

See also

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