I would like to prove or provide a counterexample to the following statement: Every group variety (i.e. an algebraic variety $X$ with a morphism $m:X\times X\to X$, $i:X\to X$, and an element $e\in X$, such that $m(m(x,y),z) = m(x,m(y,z))$, $m(i(x),x) = e$, and $m(e,x) = m(x,e)=x$ hold for all $x,y,z \in X$) is quasi-compact.
At first I thought this statement should be false, since we don't know anything about the variety except the fact that it has a group structure (which is of course very strong). But then I figured that every variety which can be covered by finitely many affine patches, is quasi-compact. So to look for a counterexample, I would have to look for a variety which consists of an infinite amount of affine pieces AND carries a group structure. I did not succeed in this.
So my thoughts were the following:
Either a group variety can be covered by finitely many affine patches (using the fact that for every subvariety $U\subset X$, $gU \cong U$ as varieties for all $g\in X$), or we might use the multiplication on an open cover. Thanks in advance for the help!