Maximal among abelian normal implies self-centralizing in nilpotent

Property-theoretic statement
In terms of group properties: the group property of being a nilpotent group is stronger than, or implies, the property of being a group in which maximal among Abelian normal implies self-centralizing.

In terms of subgroup properties: when restricting to the class of nilpotent groups, the subgroup property of being maximal among Abelian normal subgroups implies the subgroup property of being a self-centralizing subgroup.

Statement with symbols
Suppose $$G$$ is a nilpotent group and $$H$$ is maximal among Abelian normal subgroups of $$G$$: in other words, $$H$$ is an Abelian normal subgroup such that there is no Abelian normal subgroup of $$G$$ containing $$H$$. Then, $$H$$ is a self-centralizing subgroup, i.e., $$C_G(H) = H$$.

Related facts

 * Maximal among abelian normal implies self-centralizing in supersolvable: The corresponding statement holds for supersolvable groups.
 * Maximal among abelian normal not implies self-centralizing in solvable: The corresponding statement does not hold for solvable groups.
 * Maximal among abelian characteristic not implies self-centralizing in nilpotent: The corresponding statement does not hold when normality is replaced by characteristicity.
 * Thompson's critical subgroup theorem: A similar result that uses a subgroup that is maximal among Abelian characteristic subgroups.

Analogues in other algebraic structures

 * Maximal among abelian ideals implies self-centralizing in nilpotent Lie ring

Facts used

 * 1) Normality is centralizer-closed
 * 2) Normality satisfies image condition
 * 3) Nilpotence is quotient-closed: Any quotient of a nilpotent group is nilpotent.
 * 4) Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
 * 5) Cyclic over central implies Abelian
 * 6) Normality satisfies inverse image condition