Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group. 
(2) for any $g\in G$ the right-multiplication by $g$ is a morphism of algebraic varieties $G\to G$.
(3) the inverse map is a morphism $G\to G$.

Does it imply that $G$ is an algebraic group? (i.e. is the multiplication $G\times G\to G$ a morphism?)