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Symmetry Rotational symmetry group of QxQ

What is the rotational symmetry group of $\mathbb{Q}\times \mathbb{Q}$, the subset of the real plane consisting of rational points? i.e. are there infinite rotations, is this a named group ?

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Symmetry group of QxQ

What is the symmetry group of $\mathbb{Q}\times \mathbb{Q}$, the subset of the real plane consisting of rational points?