If I were to have $A$ be a finite set with order $n$ and $X\subset A$, and then define $G_X$ by:
$G_X=\{g \in$ Sym $(A) \; \vert\; g(X) = X\}$
How would I go about finding the $\vert G_X \vert$? The only thing I can think to start with is that $\vert$Sym($A)\vert$ = $n!$ But other than that, I am quite unsure.
I have already proved that $G_X \leq$ Sym($A$).
Thank you for any help.