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)
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- 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
Example of the dihedral group
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 .