Given any non abelian group, how can I prove that every proper subgroup may be abelian? I know the definition of "abelian," but I don't know the difference between a group and a subgroup, nor do I understand how the two interconnect.
Hint If $H$ is a proper subgroup of $G$ then $|H|$ is a proper divisor of $|G|$.
Hint 2 If all the proper divisors of $|G|$ are prime, then all the proper subgroups of $G$ are cyclic.
There are many non-abelian groups all of whose proper subgroups are abelian. Studying such groups of low order, we immediately find examples, such as $S_3$ or $Q_8$, the quaternion group. Because we know all subgroups explicitly for these groups, it is easy to prove that they are abelian.
One might ask what properties this class of groups has. Every such group is necessarily metabelian, for example.
More generally there is the following reference here: