Cyclic group

**roporte** · November 9th 2008, 10:15 AM

Prove that all subgroup of a cyclic group is cyclic.

Thanks!

**clic-clac** · November 9th 2008, 10:39 AM

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 $<\psi(a)>=\psi(K)$

**ThePerfectHacker** · November 9th 2008, 07:17 PM

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 $\left< a^n \right> = H$ .

Thread: Cyclic group

LinkBack

Thread Tools

Display

Cyclic group

Similar Math Help Forum Discussions

Order of Group. Direct Product of Cyclic Group

cyclic group

Cyclic Group

Prove cyclic subroups => cyclic group

automorphism group of a cyclic group

Search Tags