# Abelian normal not implies central

From Groupprops

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)

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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 .