Group in which every maximal subgroup is characteristic

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

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

• Any maximal subgroup is a characteristic subgroup.
• Any maximal subgroup is a maximal characteristic subgroup.

Relation with other properties

Stronger properties

• Cyclic group
• Group of prime power order in which every maximal subgroup is isomorph-free

Weaker properties

• Group in which every maximal subgroup is normal
• Nilpotent group for finite groups, because for a finite group, the group is nilpotent if and only if every maximal subgroup is normal. Further information: Equivalence of definitions of finite nilpotent group

Facts

• Every elementary Abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic