TITLE: Centre of the group S4
QUESTION [5 upvotes]: Quite a simple looking question guys,
Find the centre $Z(S_4)$ of $S_4$.
The previous part asked me to find centralizers for $S_4$. I note that $Id$ is the only element contained in everything so I am thinking the centre is just the trivial group {Id} but I am a little unsure how to formally prove. 
Cheers!

REPLY [2 votes]: If we denote the center of $a$ in $G$ by $C_{G}(a)$ then it holds
that 
$$
Z(G)=\cap_{a\in G}C_{G}(a)
$$
This is because the centralizer of $a$ in $G$ is the set of all
the elements the commutes with $a$ and an element is in the center
iff it commutes with all elements.
This gives justification to your reasoning that the center is trivial