Equivalence of Definitions of Conjugate of Group Element
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Theorem
The following definitions of the concept of Conjugate of Group Element are equivalent:
Definition 1
The conjugacy relation $\sim$ is defined on $G$ as:
- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$
Definition 2
The conjugacy relation $\sim$ is defined on $G$ as:
- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$
Also defined as
Some sources define the conjugate of $x$ by $a$ in $G$ as:
- $x \sim y \iff \exists a \in G: x \circ a = a \circ y$
or:
- $x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$
Proof
\(\ds a \circ x\) | \(=\) | \(\ds y \circ a\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a \circ x \circ a^{-1}\) | \(=\) | \(\ds y\) |
\(\ds x \circ a\) | \(=\) | \(\ds a \circ y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds a^{-1} \circ x \circ a\) | \(=\) | \(\ds y\) |
\(\ds \exists b \in G: \, \) | \(\ds x \circ b\) | \(=\) | \(\ds b \circ y\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds b^{-1} \circ x \circ b \circ b^{-1}\) | \(=\) | \(\ds b^{-1} \circ b \circ y \circ b^{-1}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds b^{-1} \circ x\) | \(=\) | \(\ds y \circ b^{-1}\) | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \exists a \in G: \, \) | \(\ds a \circ x\) | \(=\) | \(\ds y \circ a\) | setting $a := b^{-1}$ |
$\blacksquare$