Maximal among abelian characteristic not implies abelian of maximum order

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a group of prime power order. That is, it states that in a group of prime power order, every subgroup satisfying the first subgroup property (i.e., maximal among abelian characteristic subgroups) need not satisfy the second subgroup property (i.e., abelian subgroup of maximum order)
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Statement

It is possible to have a group of prime power order $P$, with a subgroup $K$ of $P$ that is maximal among Abelian characteristic subgroups in $P$, such that $K$ is not an Abelian subgroup of maximum order.

Related facts

• Abelian not implies contained in abelian subgroup of maximum order
• Maximal among abelian characteristic subgroups may be multiple and isomorphic
• abelian-to-normal replacement theorem for prime exponent

Proof

Example of the quaternion group

Further information: quaternion group

In the quaternion group, the center $\{ \pm 1 \}$ is the unique maximum among abelian characteristic subgroups. However, it is not an abelian subgroup of maximum order: there are cyclic subgroups of order four that are abelian.