What can I say about $a$ and $b$, that they belong to the same class when $G$ acts on $S$, same category?

Question

Suppose I have a set $S$, and a group $G$, and let $a$ and $b$ be in S, $g$ in $G$, and $g*a=b$. What can I say about $a$ and $b$, that they belong to the same class when $G$ acts on $S$, same category? How can say something like $G$ divides $S$ into equivalence classes, and $a$ and $b$ are in the same class?

Answer 1

You say that $a$ and $b$ are in the same orbit of that group action.

Yes, orbits partition the set being acted upon (in this case, $G$.)