I want to ask about Group Action.
This is the definition of group actions.
Let $G$ be group and $\Omega$ is a set. Group action $G$ on $\Omega$, defined by if there exist mapping \begin{eqnarray} \sigma : G\times \Omega &\to& \Omega\\ (g,\alpha) &\mapsto& g\alpha \end{eqnarray} and satisfies the condition
$e\alpha = \alpha$ , for all $\alpha\in\Omega$, $e$ identity element on $G$.
$(g_1g_2)\alpha= g_1(g_2\alpha)$, for all $g_1,g_2\in G$ and $\alpha \in \Omega$.
Now I want to ask about proving group actions.
Let $S_3$ be symmetric group $$S_3=\{e,(1,2),(1,3),(2,3),(1,2,3),(1,3,2)\}.$$
Let $\Omega = \mathbb{N}=\{1,2,3,\dots\}$.
Now I want to check $S_3$ acted on $\Omega$.
Let the mapping
\begin{eqnarray*} \sigma : S_3\times \mathbb{N} &\to& \mathbb{N}\\ (a,n) &\mapsto& an. \end{eqnarray*}
I check the first axiom,
- $en=n$, $\forall n\in \mathbb{N}$.
But I confused about the operation between $e=(1)(2)(3)\in S_3$ and $n\in \mathbb {N}$. How to operate element in $S_3$ and $\mathbb{N}$?