Maximal among abelian characteristic subgroups

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Definition

Symbol-free definition

A subgroup of a group is termed maximal among abelian characteristic subgroups if it is an abelian characteristic subgroup, and is not properly contained in any abelian characteristic subgroup.

Formalisms

In terms of the maximal operator

This property is obtained by applying the maximal operator to the property: Abelian characteristic subgroup
View other properties obtained by applying the maximal operator

Relation with other properties

Weaker properties

• Center of critical subgroup for a group of prime power order
• abelian characteristic subgroup
• abelian normal subgroup

Facts

Facts about such subgroups in nilpotent groups and p-groups

• Maximal among abelian characteristic not implies self-centralizing in nilpotent
• Thompson's critical subgroup theorem
• Maximal among abelian characteristic not implies abelian of maximum order

Here are some facts about the possibility of existence of multiple subgroups that are maximal among Abelian characteristic subgroups:

• Maximal among abelian characteristic subgroups may be multiple and isomorphic