Let G be cyclic and let H be a subgroup of G. Show that G/H is cyclic.

Question

Not really sure how I go about doing this. Abstract Algebra. I'm fairly certain it has to do with normal subgroups though.

Answer 1

Here is a quick answer, and if you fill out the details you'll learn a lot from it.

1) Prove that a group $G$ is cyclic if, and only if, there exists a surjective homomorphism $\mathbb Z \to G$.

2) Recall that the quotient group $G/H$ always comes equipped with the canonical surjection $\pi:G\to G/H$.

3) Combine 1 and 2.

Answer 2

$G$ is cyclic means that there is a $g \in G$ such that each $x \in G$ is of the form $x = g^k$. Now show this statement must also hold in $G/H$. The fact that $G$ is abelian might come in handy.

Answer 3

Here are some useful steps.

(1) Show that every cyclic group is abelian.

(2) Show that any subgroup $H$ of an abelian group $G$ is normal.

(3) Conclude that $G/H$ is a group.

(4) Guess a generator for $G/H$ and check that it works. (hint: consider the image of generator of $G$ under the map $G \to G/H$).