Counterexample of G-Set 
If every element of a $G$-set is left fixed by the same element $g$ of $G$,
  then $g$ must be the identity $e$.

I believe this to be true, but the answers say that it's false. Can anyone provide a counter-example? Thanks!

For concreteness, let $G$ be the group of isometries of the plane, and let $g$ be reflection in the $x$-axis. Let $S$ be the $x$-axis. Then $g(v)=v$ for every point $v$ in $S$.
