Every group is a quotient of a hypoabelian group

Statement
Let $$G$$ be a group. Then, there exists a hypoabelian group $$K$$ and a normal subgroup $$N$$ of $$K$$ such that $$G$$ is isomorphic to the quotient group $$K/N$$.

Similar facts

 * Every group is a quotient of a residually finite group
 * Every group is a quotient of a residually nilpotent group
 * Every group is a quotient of a free group

Applications

 * Hypoabelian not implies imperfect

Opposite facts

 * Nilpotency is quotient-closed: Any quotient group of a nilpotent group is a nilpotent group.
 * Solvability is quotient-closed: Any quotient group of a solvable group is a solvable group.

Facts used

 * 1) uses::Every group is a quotient of a free group
 * 2) uses::Free implies hypoabelian

Proof
The proof follows directly by combining Facts (1) and (2). More explicitly, we can take the free group that arises in the proof of Fact (1).