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>. |
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Revision as of 10:28, 11 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 .
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