Proving Normality of Subgroups in Cyclic Groups

In summary, the conversation discussed a homework question where the task was to prove that all subgroups of a group G are normal if all cyclic subgroups of G are normal. The individual asking for help mentioned their understanding of cyclic groups being abelian and their confusion about the question. The advice given was to try a proof by contradiction, where H is a subgroup of G assumed to be not normal, leading to a contradiction.
  • #1
Kuzu
15
0
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?
 
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  • #2
Try a proof by contradiction. Take H to be a subgroup of G and suppose H is not normal. What does that tell you?
 

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