Maximal among abelian characteristic subgroups

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.

Weaker properties

 * Center of critical subgroup for a group of prime power order
 * Stronger than::Abelian characteristic subgroup
 * Stronger than::Abelian normal subgroup

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