Every subgroup of an abelian group is normal, and every quotient of an abelian group is abelian. Also, a subgroup of a nonabelian group need not be normal, and a quotient of a nonabelian group need not be abelian.
Is there a simple set of (sufficient, necessary) conditions for a quotient of a nonabelian group to be abelian?
I found one here. Let $G$ be a group with commutator subgroup $G'$, let $N$ be a normal subgroup of $G$. Now $G/N$ is a abelian iff $G'$ is a subset of $N$.
Are there others?
Bonus question.
A nonabelian group where every proper subgroup is normal is called Hamiltonian.
What do you call a nonabelian group where every proper subgroup is abelian?
What do you call a nonabelian group where every quotient group is abelian? (Then the commutator subgroup is a subset of every normal subgroup.)