Maximal among abelian characteristic not implies self-centralizing in nilpotent

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a nilpotent group, every subgroup satisfying the first subgroup property need not satisfy the second subgroup property
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Statement

We can have a nilpotent group with a subgroup that is maximal among abelian characteristic subgroups (in other words, it is an abelian characteristic subgroup not contained in any bigger Abelian characteristic subgroup) that is not a self-centralizing subgroup: it is properly contained in its centralizer.

Related facts

Proof

Example of the quaternion group

Further information: quaternion group

Let Q be the quaternion group. Then, the center of Q is maximal among Abelian characteristic subgroups; however, it is far from self-centralizing, since its centralizer is the whole group.