Center is torsion-faithful in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$Z(G)$$ denotes the center of $$G$$. Then, $$Z(G)$$ is a torsion-faithful subgroup of $$G$$. In other words, if $$p$$ is a prime number such that $$Z(G)$$ is $$p$$-torsion-free, then $$G$$ is also $$p$$-torsion-free.

Dual facts
The dual fact to this is dual::center is torsion-faithful in nilpotent group.

The duality is as follows:

Facts used

 * 1) uses::Equivalence of definitions of torsion-free group for a set of primes

Proof
The proof follows from Fact (1), specifically the equivalence between (1) and (2) in Fact (1).