Everything about Conjugacy Class totally explained
In
mathematics, especially
group theory, the elements of any
group may be
partitioned into
conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. In all
abelian groups every conjugacy class is a set containing one element (
singleton set).
Functions that are constant for members of the same conjugacy class are called
class functions.
Definition
Suppose
G is a group. Two elements
a and
b of
G are called
conjugate if there exists an element
g in
G with
» gag−1 =
b.
(In
linear algebra, for
matrices this is called
similarity.)
It can be readily shown that conjugacy is an
equivalence relation and therefore partitions
G into
equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes Cl(
a) and Cl(
b) are equal
if and only if a and
b are conjugate, and
disjoint otherwise.) The equivalence class that contains the element
a in
G is
» Cl(
a) = ).
The above is particularly useful when talking about subgroups of
G. The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they're conjugate.
Conjugate subgroups are
isomorphic, but isomorphic subgroups need not be conjugate (for example, an abelian group may have two different subgroups which are isomorphic, but they're never conjugate).
Conjugacy as group action
If we define
» g.x =
gxg−1
for any two elements
g and
x in
G, then we've a
group action of
G on
G. The orbits of this action are the conjugacy classes, and the stabilizer of a given element is the element's centralizer.
Similarly, we can define a group action of
G on the set of all subsets of
G, by writing
» g.S =
gSg−1,
or on the set of the subgroups of
G.
Further Information
Get more info on 'Conjugacy Class'.
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