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 a nontrivial example, consider the action of a group $G$ on itself by conjugation, that is, let $g \circ h=g^{-1}hg$. Now $g \circ h=h$ for all $h \in G$ means that $g$ and $h$ commute for all $h \in G$, that is $g$ is in the center $Z(G)$. And $Z(G)$ need not be just the identity element.
