# Conjugacy Action on Group Elements is Group Action

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

The conjugacy action on $G$:

$\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

is a group action on itself.

## Proof

We have that:

$e * x = e \circ x \circ e^{-1} = x$

and so Group Action Axiom $\text {GA} 2$ is fulfilled.

Group Action Axiom $\text {GA} 1$ is shown to be fulfilled thus:

 $\ds \paren {g_1 \circ g_2} * x$ $=$ $\ds \paren {g_1 \circ g_2} \circ x \circ \paren {g_1 \circ g_2}^{-1}$ $\ds$ $=$ $\ds g_1 \circ g_2 \circ x \circ g_2^{-1} \circ g_1^{-1}$ $\ds$ $=$ $\ds g_1 * \paren {g_2 \circ x \circ g_2^{-1} }$ $\ds$ $=$ $\ds g_1 * \paren {g_2 * x}$

$\blacksquare$