TITLE: If G is a group such that all of its proper subgroups are abelian, then G itself must be abelian
QUESTION [5 upvotes]: Is statement below true or false?
If G is a group such that all of its proper subgroups are abelian, then G itself must be abelian

REPLY [4 votes]: To generalize @Furor's post you can consider every non-abelian Dedekind group which is called Hamiltonian Group. For infinite example of it you can consider $$Q_8\times\mathbb Z_2\times\mathbb Z(3^{\infty})$$