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>. | ||
− | { | + | == 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 23:45, 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
.