# Maximal among abelian characteristic not implies self-centralizing in nilpotent

From Groupprops

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 neednotsatisfy 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

- Maximal among abelian normal implies self-centralizing in nilpotent
- Maximal among abelian normal implies self-centralizing in supersolvable
- Nilpotent and every abelian characteristic subgroup is central implies class at most two
- Thompson's critical subgroup theorem

## Proof

### Example of the quaternion group

`Further information: quaternion group`

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