# Same rational points

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?

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Surely you want to require it be connected, or else examples from your other question will work as counterexamples. –  Charles Siegel Jul 15 '10 at 15:49

No: Take $k$ to be the rational numbers and $G$ to be the group of third roots of unity. Then the only rational point in $G$ is $1$. Then take $H$ to be the component of the identity. This satisfies your conditions but $H \neq G$.