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

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(Related facts)
 
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{{subgroup property non-implication in|
 
{{subgroup property non-implication in|
property = group of prime power order|
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group property = group of prime power order|
stronger = maximal among Abelian characteristic subgroups|
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stronger = maximal among abelian characteristic subgroups|
weaker = Abelian subgroup of maximum order}}
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weaker = abelian subgroup of maximum order}}
  
 
==Statement==
 
==Statement==
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==Related facts==
 
==Related facts==
  
* [[Abelian not implies contained in Abelian subgroup of maximum order]]
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* [[Abelian not implies contained in abelian subgroup of maximum order]]
* [[Maximal among Abelian characteristic subgroups may be multiple and isomorphic]]
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* [[Maximal among abelian characteristic subgroups may be multiple and isomorphic]]
* [[Abelian-to-normal replacement theorem for prime exponent]]
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* [[abelian-to-normal replacement theorem for prime exponent]]
  
 
==Proof==
 
==Proof==
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{{further|[[Particular example::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.
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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.

Latest revision as of 20:13, 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 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)
View all subgroup property non-implications | View all subgroup property implications

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.