Theorem: Assume that $X$ is a $G$ set.
a) If $G$ acts transitively on $X$ and $x \in X$ then $[G:stab_G(x)]=|X|$
b) If $R \subseteq X$ is a set of representatives of the G-orbits of $X$ then
$$|X|=\sum_{x \in R} [G:stab_G(x)]$$
I don't quite understand the premises
A G set is basically all the elements in the orbit of some $x$ right?. What is a G set?
a) This premise is basically saying that the number of left cosets or of St(x) equals the number of elements in the orbit of some x in the set which G acts on.
b) This is quite confusing because X is the set of elements of some orbit x, so This is saying the we can further divide up this set of elements of an orbit into more "orbit sets"
I need help with understanding this theorem.
Intuitively, orbits are those subsets of $X$ that are stable under action of $G$, which basically means that elements of $G$ map the orbit to itself. $X$ is a disjoint union of these orbits (its a short lemma).Secondly, since the orbit consists of points of form $g.x$ for all elements of $g$ of $G$ and the action of $Stab(x)$ is trivial, we mod out action of $G$ by $Stab(x)$ which results in left cosets of $Stab(x$).