# Abelian-quotient abelian normal subgroup is contained in centralizer of derived subgroup

From Groupprops

(Redirected from Abelian-quotient abelian normal subgroup is contained in centralizer of commutator subgroup)

## Statement

Suppose is a group and is an Abelian normal subgroup (?) of that is also an Abelian-quotient subgroup (?), i.e., the quotient group is an abelian group. Then, is contained in the Centralizer of commutator subgroup (?) .

## Related facts

- Commutator subgroup centralizes cyclic normal subgroup, hence any cyclic normal subgroup is contained in the centralizer of commutator subgroup.
- Commutator subgroup centralizes aut-abelian normal subgroup, hence any aut-abelian normal subgroup is contained in the 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