we know that if a group is abelian, all subgroups of it are normal. however - can you prove or disprove that if all subgroups of a group are normal, it is abelian?
We will disprove the claim.
Consider the group where . Also we need . Using the above relations you can show that is a group. You can also show it is non-abelian by observing that .
Note that is generated by any two of .
Now show that if any subgroup contains then . If , then index of in is 2 and thus is normal in . Suppose , then one of or must belong to . But then .
We can proceed similarly for the case where a subgroup contains or .
The only remaining case is that does not contain , or . In this case , which is the center of and thus is normal in