Thread: Normal Subgroups of A Group

1)If a group is abelian then all its subgroups are normal.
But turn the question around,
If all the subgroups of a group are normal then it is abelian?

I was not able to show that.
Since this is my own question I believe that is is wrong.

I think we should consider cases when the group is finite and infinite. Maybe that will help because if it is finite then it has a composition series (but I am not sure what that leads to).

Originally Posted by ThePerfectHacker

1)If a group is abelian then all its subgroups are normal.
But turn the question around,
If all the subgroups of a group are normal then it is abelian?

No; see here.

RonL

That is not exactly what I was looking for (I was hoping y conjecture to be true). But thank you.