Group in which maximal among abelian normal implies self-centralizing

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Definition

A group in which maximal among Abelian normal implies self-centralizing is a group with the following property: any subgroup that is maximal among Abelian normal subgroups is a self-centralizing subgroup.

Relation with other properties

Stronger properties

• Abelian group
• Nilpotent group: For full proof, refer: Maximal among Abelian normal implies self-centralizing in nilpotent
• Supersolvable group: For full proof, refer: Maximal among Abelian normal implies self-centralizing in supersolvable

Weaker properties

If the group is non-Abelian, then it is a group having an Abelian normal non-central subgroup