I am looking the following two questions:
1) For which natural numbers $m,n\in \mathbb{N}$ is there is there an injective group homomorphism $(\mathbb{Z}/m\mathbb{Z}, +)\rightarrow (\mathbb{Z}/n\mathbb{Z}, +)$ ?
2) For which natural numbers $m,n\in \mathbb{N}$ is there is there an surjective group homomorphism $(\mathbb{Z}/m\mathbb{Z}, +)\rightarrow (\mathbb{Z}/n\mathbb{Z}, +)$ ?
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So that the group homomorphism is injective $m$ has to be smaller than/equal to $n$. And so that the group homomorphism is surjective $n$ has to smaller than/equal to $m$.
Is my idea correct?
If such an injective homomorphism $f: \Bbb Z /m \Bbb Z \to \Bbb Z / n\Bbb Z$ exists, then $f(\Bbb Z/m\Bbb Z)$ forms a subgroup in $\Bbb Z/ n \Bbb Z$, now use Lagrange’s theorem.
For surjective homomorphism, for each element in $\Bbb Z/ n\Bbb Z$ pick one of its preimage, then there is an isomorphism between the codomain and a subset of $\Bbb Z/m\Bbb Z$. Then Lagrange’s theorem.
When the condition above are satisfied, you can explicitly construct required homomorphisms, hence the condition is sufficient and necessary.