All subgroups normal $\implies$ abelian group This is , I think an easy problem just that I am not getting the catch of it. How to show whether or not the statement is true?
All subgroups of a group are normal$\implies$ the group is an abelian group?
I have been able to show the other way round.

This is actually not true. A group for which all subgroups are normal is called a Dedekind group, and non-abelian ones are called "Hamiltonian". The smallest example is the quaternion group $Q_8$. See this MO discussion for more info.
