Derived subgroup centralizes cyclic normal subgroup

Statement
Suppose $$N$$ is a cyclic normal subgroup of a group $$G$$. Then, the derived subgroup $$[G,G]$$ is contained in the centralizer $$C_G(N)$$.

Equivalently, since centralizing is a symmetric relation, we can say that $$N$$ is contained in the centralizer of derived subgroup $$C_G([G,G])$$.

Related facts about cyclic normal subgroups

 * Normal of least prime order implies central
 * Cyclic normal Sylow subgroup for least prime divisor is central

Related facts about descent of action

 * Quotient group acts on abelian normal subgroup

Related facts about containment in the centralizer of commutator subgroup

 * Derived subgroup centralizes aut-abelian normal subgroup, so any aut-abelian normal subgroup is contained in the centralizer of derived subgroup
 * Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup
 * Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group
 * Abelian subgroup equals centralizer of derived subgroup in generalized dihedral group unless it is a 2-group of exponent at most four

Other related facts

 * Odd-order cyclic group is characteristic in holomorph

Facts used

 * 1) uses::Cyclic implies aut-abelian
 * 2) uses::Derived subgroup centralizes aut-abelian normal subgroup

Proof
The proof follows from facts (1) and (2).