Difference between revisions of "Centralizer"
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If <math>X \subseteq Y</math> are subsets of a magma <math>E</math>, then <math>Y' \subseteq X'</math>. The ''bicentralizer'' <math>X''</math> of <math>X</math> is the centralizer of <math>X'</math>. Evidently, <math>X \subseteq X''</math>. The centralizer of the bicentralizer, <math>X'''</math>, is equal to <math>X'</math>, for <math>X' \subseteq X'''</math>, but <math>X \subseteq X''</math>, so <math>X''' \subseteq X'</math>. | If <math>X \subseteq Y</math> are subsets of a magma <math>E</math>, then <math>Y' \subseteq X'</math>. The ''bicentralizer'' <math>X''</math> of <math>X</math> is the centralizer of <math>X'</math>. Evidently, <math>X \subseteq X''</math>. The centralizer of the bicentralizer, <math>X'''</math>, is equal to <math>X'</math>, for <math>X' \subseteq X'''</math>, but <math>X \subseteq X''</math>, so <math>X''' \subseteq X'</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]]'' 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>. |
| − | { | + | == Centralizers in Groups == |
| + | |||
| + | 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)]] | ||
| + | * [[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 
be a magma, and let 
be a subset of . The centralizer 
of is the set of elements of which commute with every element of .
If 
are subsets of a magma , then 
. The bicentralizer 
of is the centralizer of . Evidently, 
. The centralizer of the bicentralizer, 
, is equal to , for 
, but , so 
.
If the magma is associative, then the centralizer of is also the centralizer of the subset of genererated by , and the centralizer of is furthermore an associative sub-magma of . If is a group, then the centralizer of is a subgroup, though not necessarily normal. The centralizer of is also called the center of .
Centralizers in Groups
If 
is a group, then an element 
of is said to centralize 
if it commutes with every element of ; that is, if 
for all 
. A subset 
of is said to centralize if all its elements centralize . The centralizer of , denoted 
, or 
when there is no risk of confusion, is the set of elements that centralize . It is evidently a subgroup of .
