I would like to show that any Zariski-closed subsemigroup of $SL_n(\mathbb{C})$ is a group. If I understand correctly, this is consequence 1.2.A of http://www.heldermann-verlag.de/jlt/jlt03/BOSLAT.PDF .
Is there a more elementary proof? For $SL_2(\mathbb{C})$, the result is quite easy to show directly, or using the Hilbert basis theorem, .