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There does not exist any nontrivial group homomorphism from $(\mathbb{Q},+)$ to $(\mathbb{Q}^*,\cdot)$.
Is it also true that there does not exist any nontrivial group homomorphism from $(\mathbb{Q}^*,\cdot)$ to $(\mathbb{Q},+)$?
If it is not true then how can we find all those nontrivial homomo...