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### Problem 8

Let $G$ be a group, $h \in G$ and $n \in \mathbb{Z}^{+}$. Let $\phi: \mathbb{Z}_n \rightarrow G$ be defined by $\phi(i) = h^{i}$ for $0 <= i <= n$. Give a necessary and sufficient condition (in terms of $h$ and $n$) for $\phi$ to be a homomorphism. Prove your assertion.

**Solution**

The map $\phi$ is a homomorphism if and only if $h ^ n = e$ where $e$ is the identity of $G$.

(=>) If $\phi$ is a homomorphism then $e = \phi(0) = \phi(1)^n = h ^ n$.

(<=) Suppose that $h^n = e$, so that $\langle h \rangle \cong Z_{m}$ where $m$ is a divisor of $n$, and let $i, j \in \mathbb{Z}_n$. By the division algorithm, $\phi(i+j) = \phi(i)\phi(j)$, and then $\phi$ is a homomorphism.