Center is quotient-torsion-freeness-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$p$$ is a prime number such that $$G$$ is $$p$$-torsion-free. Suppose $$Z(G)$$ is the center of $$G$$. Then the quotient group $$G/Z(G)$$, which is also isomorphic to the inner automorphism group of $$G$$, is also $$p$$-torsion-free.

In other words, the center of $$G$$ is a quotient-torsion-freeness-closed subgroup of $$G$$.

Dual fact
The dual fact to this is derived subgroup is divisibility-closed in nilpotent group.

Facts used

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

Proof
The proof follows directly from Fact (1), specifically the (2) implies (6) implication, setting $$i = c, j = 1$$ where $$c$$ is the nilpotency class of $$G$$.