Given the natural coalgebra structure on a group algebra $kG$, one can recover the group by taking the set of group-like elements of the coalgebra $kG$.
When can you go the other way? In particular, given a Hopf algebra $H$, under what conditions can one recover the structure of $H$ from it's group of group-like elements?
I'm also curious as to how the answer differs if $H$ is finitely generated versus finite dimensional.
Thanks!