Group in which every abelian characteristic subgroup is central

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Definition

Symbol-free definition

A group in which every Abelian characteristic subgroup is central is a group satisfying the following equivalent conditions:

• Every Abelian characteristic subgroup of the group is central: it is contained in the center.
• The center is maximal among Abelian characteristic subgroups: there is no Abelian characteristic subgroup properly containing the center.

Relation with other properties

Stronger properties

• Abelian group
• Simple group
• Characteristically simple group
• Fitting-free group
• Group in which every Abelian normal subgroup is central

Related subgroup properties

• Constructibly critical subgroup: A critical subgroup arising via the constructive procedure of Thompson's critical subgroup theorem automatically satisfies this condition: its center is the unique maximum among Abelian characteristic subgroups.