If $G$ is a cyclic group of order $d$ with generator $a$, show that $\dfrac{\mathbb{Z}}{d\mathbb{Z}}$ is isomorphic $G$.

Question

We have to $|G|=d$ and $G=\left<a\right>=\{a^n : n \in \mathbb{Z}\}$ by definition of cyclic group.

To show what you are asking, I want to define a function, not necessarily this $$\phi : \mathbb{Z} \to G \text{ such that } z \mapsto \phi(z)=a^d$$ and use the theorem: Let $\varphi:G \to G'$ be a group homomorphism, then $$\dfrac{G}{ker(\varphi)} \simeq Im(\varphi)$$

I need your help to define well a function that serves me.

Answer 1

Hint: Define $\phi (x)=a^x$, for each $x\in\mathbb Z$.