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What are all homomorphisms from $\mathbb Z_n$ to $\mathbb Z_n$ ? I know about all automorphisms but am not clear about all homomorphisms ; are there a total of $n$ homomorphisms ? In general , what are all homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ?

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  • $\begingroup$ Hints: (1) Since $\mathbb{Z}_m$ is generated by $1$, any homomorphism is determined by its action on $1$. (2) Any homomorphism $\phi$ must satisfy $m\phi(1) = \phi(m) = \phi(0) = 0$. $\endgroup$ – Travis Willse Sep 13 '14 at 14:05
  • $\begingroup$ An homomorphism $f: \mathbb{Z}_m \longrightarrow \mathbb{Z}_n$ is uniquely determined by $f(1)$. So all you need is to ensure that $m \cdot f(1) = 0$. $\endgroup$ – Crostul Sep 13 '14 at 14:06