Derived subgroup centralizes cyclic normal subgroup

Statement

Suppose $N$ is a Cyclic normal subgroup (?) of a group $G$. Then, the commutator 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 commutator subgroup (?) $C_G([G,G])$.

Related facts

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

Facts used

1. Cyclic implies aut-abelian
2. Commutator subgroup centralizes aut-abelian normal subgroup

Proof

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