Definition:Conjugate (Group Theory)/Element

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Definition

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

An element $x \in G$ is conjugate to an element $y \in G$ iff:

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

Alternatively, we can say that $x$ is the conjugate of $y$ by $a$.

This relation is called conjugacy.

We write $x \sim y$ for $x$ is a conjugate of $y$.

This relation is alternatively (and usually) expressed as:

$x \sim y := a \circ x \circ a^{-1} = y$

which is seen to be equivalent to the other definition by taking the group product on the right with $a^{-1}$.

There is an alternative way of defining conjugacy of elements, which is subtly different:

$x$ is a conjugate of $y$ iff:

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

or:

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

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

Some sources call $a \circ x \circ a^{-1}$ (or $a^{-1} \circ x \circ a$) the transform of $x$ by $a$.