Let $X$ be a set, $G$ the group of bijection on $X$. Then it is well-known that if $|X|\geq 3$, $Z(G)$ is trivial. However, I cannot see a way of extending this proof to $X$ being an infinite set (other than when a bijection only moves finitely many points). Indeed, I have been somewhat stumped trying to prove that $Z(G)$ is or is not trivial for $X$ infinite.

So, for $X$ an infinite set, $G$ the bijections on $X$, is it true that $Z(G)$ is trivial?