Abelian normal not implies central

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) need not satisfy 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
Get more facts about abelian normal subgroup|Get more facts about central subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property abelian normal subgroup but not central subgroup|View examples of subgroups satisfying property abelian normal subgroup and central subgroup

Statement

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

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

• Normal subgroup of least prime order is central
• Abelian normal implies commutator-in-center

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 $G$ to be the dihedral group of order eight, and $H$ to be any of the three maximal subgroups of $G$, then $H$ is abelian and normal in $G$ but is not central in $G$.