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

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

• Commutator subgroup centralizes aut-abelian normal subgroup, so any aut-abelian normal subgroup is contained in the centralizer of commutator subgroup
• Abelian-quotient abelian normal subgroup is contained in centralizer of commutator subgroup
• Abelian subgroup is contained in centralizer of commutator subgroup in generalized dihedral group
• Abelian subgroup equals centralizer of commutator 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. Cyclic implies aut-abelian
2. Derived subgroup centralizes aut-abelian normal subgroup

Proof

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