If $G$ is a Lie group with $Z(G)=\{g\in G:hg=gh \;\;\;\text{for all $h\in H$}\}=\{e\}$
Then is $G$ necessarily connected?
I tried to find some counterexamples to no avail; it seems quite hard to find centerless Lie groups in the first place.
On the other hand, I think $G$ might be path-connected through paths to $e$.
Any helps are appreciated!
No: Take any centerless connected Lie group (e.g. $\mathrm{SO}(3)$), any centerless finite group (e.g. $S_3$) and take their cartesian product. So for example, $$\mathrm{SO}(3)\times S_3$$ has no centre but six connected components.