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) need not satisfy the second subgroup property (i.e., central subgroup)
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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

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 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.