Conjugacy is Equivalence Relation

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Theorem

Conjugacy of group elements is an equivalence relation.


Proof

Checking each of the criteria for an equivalence relation in turn:


Reflexive

$\forall x \in G: e_G \circ x = x \circ e_G \implies x \sim x$

Thus conjugacy of group elements is reflexive.

$\Box$


Symmetric

\(\displaystyle \) \(\) \(\displaystyle x \sim y\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle a \circ x = y \circ a\) Definition of Conjugate of Group Element
\(\displaystyle \) \(\leadsto\) \(\displaystyle a \circ x \circ a^{-1} = y\) Definition of Group
\(\displaystyle \) \(\leadsto\) \(\displaystyle a^{-1} \circ y = x \circ a^{-1}\) Definition of Group
\(\displaystyle \) \(\leadsto\) \(\displaystyle y \sim x\) Definition of Conjugate of Group Element

Thus conjugacy of group elements is symmetric.

$\Box$


Transitive

\(\displaystyle \) \(\) \(\displaystyle x \sim y, y \sim z\)
\(\displaystyle \) \(\leadsto\) \(\displaystyle a_1 \circ x = y \circ a_1, a_2 \circ y = z \circ a_2\) Definition of Conjugate of Group Element
\(\displaystyle \) \(\leadsto\) \(\displaystyle a_2 \circ a_1 \circ x = a_2 \circ y \circ a_1 = z \circ a_2 \circ a_1\) Definition of Group
\(\displaystyle \) \(\leadsto\) \(\displaystyle x \sim z\) Definition of Conjugate of Group Element

Thus conjugacy of group elements is transitive.

$\Box$


All criteria are satisfied, and so conjugacy of group elements is shown to be an equivalence relation.

$\blacksquare$


Sources