Definition:Group Action

Definition 1
Let $X$ be a set.

Let $G$ be a group whose identity is $e$.

A group action is a mapping $\phi: G \times X \to X$ such that:


 * $\forall \left({g, x}\right) \in G \times X: g * x := \phi \left({\left({g, x}\right)}\right) \in X$

in such a way that:
 * GA-1: $\forall g, h \in G, x \in X: g * \left({h * x}\right) = \left({g h}\right) * x$;
 * GA-2: $\forall x \in X: e * x = x$.

We say that the group $G$ acts on the set $X$.

Definition 2
Let $X$ be a set, and let $\Gamma \left({X}\right)$ be the group of permutations of $X$.

Let $G$ be a group whose identity is $e$.

A group action is a group homomorphism $\phi: G \to \Gamma \left({X}\right), g \mapsto \phi_g$.

That is, the following holds:

One says that $G$ acts on $X$.

Also known as
Some sources call this a $G$-set.

Some sources use $g \wedge x$ for $g * x$, while some use $g \cdot x$.

Some sources introduce the concept with the notation $\phi_g \left({x}\right)$ for $g * x$, before progressing to the latter notation.

There is little consistency in the literature; $*$ appears to be popular. $\wedge$ is not generally preferred, because its other uses are somewhat specialized.

Also see

 * Equivalence of Definitions of Group Action, proving the definitions presented above are equivalent.