# On quotient spaces of affine algebraic groups by conjugate subgroups

Let $$H$$ be a closed, connected subgroup of an affine algebraic group $$G$$.

Then is it true that for every $$x\in G$$, the quotient spaces $$G/H$$ and $$G/xHx^{-1}$$ are isomorphic as varieties ?

Where I am considering $$G/H$$ as a variety in the usual sense as described in Humphreys, Linear Algebraic Groups, Chapter IV

• Make your last sentence concrete.. – reuns Apr 26 '19 at 1:17
• reuns: I think the variety stricture on $G/H$ is very standard ... I have seen it in every book on algebraic groups I've come across so far ... – user102248 Apr 26 '19 at 1:22

By the universal property of quotients of algebraic groups, the composition $$G \xrightarrow{\operatorname{Int} x} G \rightarrow G/xHx^{-1}$$ induces a morphism of varieties $$G/H \rightarrow G/xHx^{-1}$$. Similarly you can obtain a morphism $$G/xHx^{-1} \rightarrow G/H$$ and check that these are inverse to each other.