Maximal among abelian characteristic not implies abelian of maximum order

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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 [[{{{group property}}}]]. That is, it states that in a [[{{{group property}}}]], 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

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.