Let $G$ be a connected nilpotent Lie group.
Firstly, I would like to know whether it is true that $G$ is simply connected if and only if its center is.
I do not understand how the center of $G$ being simply-connected can imply that $G$ is.
Secondly, I would like to know how to show that if $G$ is simply-connected,then $exp:Lie(G)\rightarrow G$ is a diffeomorphism. I know that it is surjective and that $G$ is contractible, but, so far, I've been unable to finish the argument.