Refer to Lang's Algebra p. 80 exercise 46. Let $G$ be a finite group acting on a finite set $S$. Then Lang calls a partition $S = \bigcup_{i \in I} S_i$ of $S$ "stable" if $G$ maps each $S_i$ onto some $S_j$.
Does that mean that for any $i \in I$ and for any $x \in G$ there exists $j \in I$ such that $x S_i=S_j$ or that for any $i \in I$ there exists $j \in I$ such that $x S_i = S_j$ for all $x \in G$? Notice that in the first interpretation $j$ depends on $x$, while in the second interpretation it does not. Which of the two is the correct interpretation?
Thanks.