All subgroups normal $\implies$ abelian group

normal-subgroups abelian-groups

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.


Hint: consider $Q_{8}$, the quaternion group.

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