Every group is a quotient of a residually nilpotent group

This article gives the statement, and possibly proof, of a group property (i.e., residually nilpotent 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 residually nilpotent 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 free group

Applications

• Every group is a quotient of a hypoabelian group
• Residually nilpotent not implies imperfect

Opposite facts

• Nilpotency is quotient-closed: Any quotient group of a nilpotent group is a nilpotent group.

Facts used

1. Every group is a quotient of a free group
2. Free implies residually nilpotent

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).