Nontrivial subgroup of nilpotent group has nontrivial homomorphism to center

Statement

Suppose $G$ is a nontrivial nilpotent group. Denote by $Z(G)$ the center of $G$. Suppose $H$ is a nontrivial subgroup of $G$. Then, there exists a nontrivial homomorphism of groups $\varphi:H \to Z(G)$.

Related facts

• Nilpotent implies center is normality-large
• Equivalence of definitions of characteristic direct factor of nilpotent group

Proof

The idea is to use an iterated commutator operation where all the other coordinates are fixed.