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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

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  • $\begingroup$ Make your last sentence concrete.. $\endgroup$
    – reuns
    Apr 26, 2019 at 1:17
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    $\begingroup$ 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 ... $\endgroup$
    – user102248
    Apr 26, 2019 at 1:22

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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.

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