Definition:Conjugate (Group Theory)/Element/Also defined as

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Let $\left({G, \circ}\right)$ be a group.

Some sources define the conjugate of $x$ by $a$ in $G$ as:

$x \sim y \iff \exists a \in G: x \circ a = a \circ y$


$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$

This is clearly equivalent to the other definition by noting that if $a \in G$ then $a^{-1} \in G$ also.

Also known as

Some sources refer to the conjugate of $x$ as the transform of $x$.

Some sources refer to conjugacy as conjugation.

Also see

  • Results about conjugacy can be found here.