This note conssits of two rather separated parts. In the first part (§1), we remark a property of homomorphisms of group varieties, and in the second part (§2 and the following), we prove that if a group variety $G$ contains a group subvariety $H$, there exists a non-singular variety $V$ which has $G$ as a group of transformations, and whose points are in one-to-one correspondence with the cosets of $H$ in $G$.