Group Action is a Homomorphism

Theorem

Let $\Gamma \left({X}\right)$ be the set of permutations on a set $X$.

Let $G$ be a group.

A group action is a (group) homomorphism from $G$ to $\Gamma \left({X}\right)$.

Proof

Let $g, h \in G$.

From the definition of group action, $\forall \left({g, x}\right) \in G \times X: \phi \left({\left({g, x}\right)}\right) \in X = g \wedge x \in X$.

Let $\phi_g: X \to X$ be the mapping defined as $\phi_g \left({x}\right) = \phi \left({g, x}\right)$.

Let $\phi \left({g, x}\right) = \phi_g \left({x}\right)$.

• First we show that $\phi_g \circ \phi_h \left({x}\right) = \phi_g \phi_h \left({x}\right)$.

	\(\displaystyle \)	\(\displaystyle \phi_g \circ \phi_h \left({x}\right)\)	\(=\)	\(\displaystyle g \wedge \left({h \wedge x}\right)\)	\(\displaystyle \)	by definition of $\phi_g$, $\phi_h$
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({g h}\right) \wedge x\)	\(\displaystyle \)	by definition of group action
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi_g \phi_h \left({x}\right)\)	\(\displaystyle \)

• We have:

$e \wedge x = x \implies \phi_e \left({x}\right) = x$

where $e$ is the identity of $G$.

$\blacksquare$

Comment

Some treatments of this subject take this as the definition of a group action.