conjugacy class

Noun

Meaning

conjugacy class (plural conjugacy classes)

(algebra) A subset of a group which is an equivalence class in the quotient set of the group divided by conjugation as equivalence relation.
For a given element



x


{\displaystyle x}

of some group



G


{\displaystyle G}

, its conjugacy class




x

G




{\displaystyle x^{G}}

is



{
y
∈
G

|

∃
g
∈
G
:

g

−
1


x
g
=
y
}


{\displaystyle \{y\in G|\exists g\in G:g^{-1}xg=y\}}

.
If a group acts on itself through conjugation, then its orbits are its conjugacy classes and its stabilizer subgroups are its centralizers.

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