# 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) neednotsatisfy 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.