# Abelian subgroup is contained in centralizer of derived subgroup in generalized dihedral group

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(Redirected from Abelian subgroup is contained in centralizer of commutator subgroup in generalized dihedral group)

## Contents

## Statement

Suppose is an abelian group and is the corresponding Generalized dihedral group (?) containing as an abelian normal subgroup of index two. Then, is contained in the Centralizer of commutator subgroup (?) . In other words, the commutator subgroup centralizes .

## Related facts

### Related facts about generalized dihedral groups

- Abelian subgroup is isomorph-containing in generalized dihedral group unless it is an elementary abelian 2-group
- Abelian subgroup equals centralizer of commutator subgroup in generalized dihedral group unless it is a 2-group of exponent at most four

### Other facts about being contained in the centralizer of commutator subgroup

- Commutator subgroup centralizes cyclic normal subgroup, or equivalently, any cyclic normal subgroup of a group is contained in the centralizer of commutator subgroup.
- Commutator subgroup centralizes aut-abelian normal subgroup, or equivalently, any aut-abelian normal subgroup of a group is contained in the centralizer of commutator subgroup.
- Abelian-quotient abelian normal subgroup is contained in centralizer of commutator subgroup