Nontrivial subgroup of nilpotent group has nontrivial homomorphism to center

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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

Proof

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