Proving Normality of Subgroups in Cyclic Groups

[Kuzu]

I'm taking this course "abstract algebra" at university and I've been given some homework questions. I was able to solve all of them but one. And it would be great if anyone could help me with this.

The question is like this:
"If all cyclic subgroups of G are normal, then show that all subgroups of G are normal"


as I know all cyclic groups are abelian, and G itself is a subgroup of G so it is cyclic and abelian. Also I know that every subgroup of an abelian group is normal. So I didn't even understand the question completely.

How can I solve this?

 

[Dick]

Try a proof by contradiction. Take H to be a subgroup of G and suppose H is not normal. What does that tell you?