Notation for permutation corresponding to the action of a group element

Question

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e.,

1. $e.x = x$ for all $x \in X$;
2. $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$.

For a fixed $g \in G$, how should I refer to the permutation $g.\cdot : X \to X$? (I.e., the map $(x \mapsto g.x) : X \to X$.) Having the $.$ right next to a $\cdot$ is ugly to me. Is there a nicer notation for this, or perhaps a standard English phrase to refer to that permutation?

Answer 1

Let me have short review.

Some books use $`` x^g "$ like Isaac's Group theory, while others use $gx$.

For every action, we have permutation representation. Assume that $G$ acts on set $X$. A permutation representation is a homomorphism $\phi$ between $G$ and $S_X$ where by $S_X$ we mean symmetric group on $X$. We can define $\phi$ by helping from action $G$ on $X$, i.e. $\phi(g)=\phi_g(x)=x^g$.

For instance, Let $G$ be a group with a proper and non-trivial subgroup $H$. Assume that $X$ is set of right cosets. So as you know you$G$ acts on $X$ and $(Hx)^g=Hxg$.

Now, we can talk about permutation representation of this action. Thus, we have $\phi:G\rightarrow S_X$ such that $\phi_g(Hx)=(Hx)^g=Hxg$.