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, .



Zariski-closed subsemigroups of SL_n(C) are groups
It is quite elementary. Let $S$ be the semi-group in question. Then for any $g \in S$, the set
$g^kS$ for $k=1,2, \dots$ is a decreasing sequence of closed sets, hence it has to stabilize. So, $g^kS=g^{k+1}S$ implies that $gS=S$. Hence $S$ is closed with respect to taking inverse, and therefore is a group. 


