1. ## Cyclic group

Prove that all subgroup of a cyclic group is cyclic.

Thanks!

2. Let $(G,.)$ be a cyclic group, $H$ a subgroup of $G$, and $\alpha \in G$ such that $<\alpha >=G$.
Let $\phi : \mathbb{Z} \rightarrow G : n \mapsto \alpha^{n}$ be the only morphism of groups such that $\phi(1)=\alpha$.

Then $\phi^{-1}(H)$ is a subgroup of $(\mathbb{Z},+)$, so it's a cyclic group.
So $\phi(\phi^{-1}(H))=H$ (because of surjectivity) is a cyclic group.

Why, if $K$ is a cyclic group, $L$ another group, and $\psi$ a morphism of groups between $K$ and $L$, then $\psi(K)$ is cyclic?

Let $a$ be a generator of $K$, and $b$ an element of $\psi(K)$. There is a $n \in \mathbb{N}$ such that $b=\psi(a^{n})=\psi(a)^{n}$. Thus $<\psi(a)>=\psi(K)$

3. Originally Posted by roporte
Prove that all subgroup of a cyclic group is cyclic.

Thanks!
Let $H$ be a subgroup. And $a$ generate the large group $G$. Choose $n\in \mathbb{Z}^{\times}$ to be the smallest such that $a^n\in H$. Now prove that $\left< a^n \right> = H$.