View source for Maximal among abelian characteristic not implies abelian of maximum order

{{subgroup property non-implication in|
group property = group of prime power order|
stronger = maximal among abelian characteristic subgroups|
weaker = abelian subgroup of maximum order}}

==Statement==

It is possible to have a [[group of prime power order]] <math>P</math>, with a subgroup <math>K</math> of <math>P</math> that is [[maximal among Abelian characteristic subgroups]] in <math>P</math>, such that <math>K</math> 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|[[Particular example::quaternion group]]}}

In the quaternion group, the center <math>\{ \pm 1 \}</math> 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.