In Springer's 'Linear Algebraic Groups', he proves that if $G$ is a linear (=affine) algebraic group and $H$ is a closed normal subgroup of $G$, then $G/H$ has a linear algebraic group structure with the usual group structure. But I think there's some problem with the proof:
I think there's a problem with the third line, since the map $(xH, yH)\mapsto xy^{-1}H$ doesn't give the usual group structure on $G/H$. I think it is okay with the latter part, so I want to know whether the first paragraph of the proof is right or wrong. Thanks in advance.
It doesn't matter because he's just trying to show that the variety is affine. The group structure is unimportant for this part of the argument.