# Difference between revisions of "Maximal among abelian characteristic not implies abelian of maximum order"

From Groupprops

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{{subgroup property non-implication in| | {{subgroup property non-implication in| | ||

property = group of prime power order| | property = group of prime power order| | ||

− | stronger = maximal among | + | stronger = maximal among abelian characteristic subgroups| |

− | weaker = | + | weaker = abelian subgroup of maximum order}} |

==Statement== | ==Statement== |

## Revision as of 20:01, 3 February 2009

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) neednotsatisfy the second subgroup property (i.e., abelian subgroup of maximum order)

View all subgroup property non-implications | View all subgroup property implications

## 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.