Every group is a quotient of a hypoabelian group

This article gives the statement, and possibly proof, of a group property (i.e., hypoabelian group) satisfying a group metaproperty (i.e., quotient-universal group property)
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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 $K/N$.

Related facts

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. Every group is a quotient of a free group
2. 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).