# Derived subgroup centralizes cyclic normal subgroup

From Groupprops

## Contents

## Statement

Suppose is a cyclic normal subgroup of a group . Then, the derived subgroup is contained in the centralizer .

Equivalently, since centralizing is a symmetric relation, we can say that is contained in the centralizer of derived subgroup .

## 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

### 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

## Facts used

## Proof

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