Let $G=G\_1\times G\_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the category of varieties and is a $G$-torsor locally trivial in the etale topology. 


Is it true, that $X \rightarrow X/G\_1$ exists in the category of varieties? 


Is it an etale locally trivial $G\_1$-torsor?


Is $X/G\_1\rightarrow X/G$ an etale locally trivial $G\_2$-torsor?


If not does it help to assume the $G\_i$ to be reductive?


What if instead $G\_1$ is only a normal subgoup of $G$ and $G\_2$ the quotient group?



Existence of quotient variety for group implies existence of quotient for normal subgroups
In what follows, we only need $G\_1$ to be normal in $G$, as in the last question. Put $Y=X/G$, and $Z=X/G\_1$: $Y$ is a variety by assumption, and both $X$ and $Z$ make sense as sheaves on the étale site of $Y$. On this site, $X\to Y$ is locally isomorphic to $Y\times G$ with the obvious action of $G$; it follows that $Z$ is locally isomorphic to $Y\times G\_2$, i.e. $Z$ is a $G\_2$-torsor over $Y$. But $G\_2$ is affine by assumption. So, by flat descent of (quasi-)affine schemes (SGA 1, VIII, 7.9), $Z$ is a scheme, affine over $Y$. Moreover, $X\to Z$ is locally over $Z$ (even over $Y$) isomorphic to $Z\times G\_1\to Z$, so it is a $G\_1$-torsor. 


Everything works just as well with "étale" replaced by "fppf". 


Note that the sheaf-theoretic arguments here are completely formal. The key assumption for the representability of $X/G\_1$ by a scheme is the fact that $G\_2$ is affine. For more general algebraic groups, the "right" viewpoint is to forget about varieties and work with algebraic spaces: $Z$ is always an algebraic space and $X\to Z$ (resp. $Z\to Y$) is a $G\_{1,Z}$-torsor (resp. a $G\_{2,Y}$-torsor).


