How many group homomorphisms are there from $(\mathbb{Q}^+, \cdot\ )$ to $(\mathbb{Z}_m,+)$?
Of course, there is a trivial zero homomorphism. I think there are no more homomorphisms there but I am failing to prove it.
2026-03-26 23:09:30.1774566570
How many group homomorphisms are there from $(\mathbb{Q}^+, \cdot\ )$ to $(\mathbb{Z}_m,+)$
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Hint: The fundamental theorem of arithmetic says that $(\mathbb{Q}^+, \cdot\ ) \cong \prod_p (\mathbb Z,+)$, the product being over the prime numbers. Therefore, the question reduces to finding all group homomorphisms from $(\mathbb{Z},+)$ to $(\mathbb{Z}_m,+)$, which is much easier.