# Derived subgroup centralizes normal subgroup whose automorphism group is abelian

## Contents

## Statement

Suppose is a normal subgroup whose automorphism group is abelian (i.e., a normal subgroup that is a group whose automorphism group is abelian -- it has an abelian automorphism group) 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

- Commutator subgroup centralizes cyclic normal subgroup
- 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

- Commutator subgroup centralizes cyclic normal subgroup, so any cyclic 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

## Proof

**Given**: A group . A cyclic normal subgroup .

**To prove**: .

**Proof**: Consider the homomorphism:

given by:

.

Note that this map is well-defined because is normal in , so gives an automorphism of for any .

- The kernel of is : This is by definition of centralizer: is the set of such that for all , which is equivalent to being in the kernel of .
- The kernel of contains : Since is a homomorphism to an abelian group, is the identity. Thus, every commutator lies in the kernel of , so is in the kernel of .
- : This follows by combining steps (1) and (2).