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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Example of the quaternion group
Further information: quaternion group
In the quaternion group, the center 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.