Proof
(related to Theorem: Classification of Cyclic Groups)
- By hypothesis, $(G,\ast)$ is a cyclic group with a generator $g\in G$ with $G=\langle g\rangle.$
- Define the function $f:(\mathbb Z, + )\to (G,\ast),$ by $f(k):=a^k$ for all $a\in (G,\ast)$ and all elements $k$ of the additive group of integers $k\in (\mathbb Z,+).$
- $f$ is a group homomorphism, since $f(k+j)=a^{k+j}=a^k\ast a^j=f(k)\ast f(j).$
- In particular, the image $\operatorname{im}(f)$ equals the group $G$, since $\operatorname{im}(f)=\{a^k\mid k\in\mathbb Z\}=\langle g\rangle=G.$
- On the other hand, the kernel $\ker(f)=\{a\in G\mid f(a)=e_G\}$ is a subgroup of $(Z,+)$.
- Let $G$ have an infinite order $|G|=\infty.$
- Let $G$ have an finite order $|G|=n$ with $n > 0.$
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