# Abelian normal not implies central

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., abelian normal subgroup) neednotsatisfy the second subgroup property (i.e., central subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

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## Contents

## Statement

It is possible to have a group and an abelian normal subgroup of (i.e., is an abelian group and is a normal subgroup of ) that is not a central subgroup of (i.e., is not contained in the center of ).

## Related facts

### Similar facts

- Maximal among abelian normal implies self-centralizing in nilpotent: This shows that in a nilpotent group and in particular in a group of prime power order, any subgroup that is maximal among abelian normal subgroups is a self-centralizing subgroup. In particular, if the whole group is not abelian, it cannot be a central subgroup. Examples include dihedral group:D8, quaternion group, and many others.
- Maximal among abelian normal implies self-centralizing in supersolvable: The result also holds for supersolvable groups, such as the symmetric group of degree three.
- Normal not implies central factor
- Abelian-quotient not implies cocentral
- Nilpotent and every abelian characteristic subgroup is central implies class at most two

### Opposite facts

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, Klein four-subgroups of dihedral group:D8`

If we take to be the dihedral group of order eight, and to be any of the three maximal subgroups of , then is abelian and normal in but is not central in .