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Let G and H be affine algebraic groups defined over a field k of characteristic zero, with H a closed subgroup of G. Suppose they have the same k-points. Have they to be equal?

Let $G$ be an affine reductive group defined over a field of characteristic zero $k$, denote by $\bar{k}$ the algebraic closure of $k$, and by $k^{\times}$ the multiplicative group of $k$. Let $Z(G)$ …

Let $G$ be an affine algebraic group defined over a field of characteristic zero $K$. Suppose $G$ has only one single $K$-point, can we conclude that $G$ does not have more points?

Let $G$ be a reductive algebraic group defined over a field $k$ and $X$ an affine $G$-variety. In the case $k$ is algebraically closed we have the following result: Let $x\in X$ such that the orbit …