Maximal among abelian characteristic not implies abelian of maximum order
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 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 , with a subgroup
of
that is maximal among Abelian characteristic subgroups in
, such that
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 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.