Class two not implies abelian automorphism group: Difference between revisions
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==Related facts== | ==Related facts== | ||
* [[Cyclic implies | * [[Cyclic implies abelian automorphism group]] | ||
* [[ | * [[Abelian automorphism group implies class two]] | ||
* [[ | * [[Abelian automorphism group not implies abelian]] | ||
===Stronger facts=== | ===Stronger facts=== | ||
* [[Weaker than::Finite abelian and | * [[Weaker than::Finite abelian and abelian automorphism group implies cyclic]] | ||
==Proof== | ==Proof== | ||
Revision as of 19:34, 20 June 2013
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group of nilpotency class two) need not satisfy the second group property (i.e., aut-abelian group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about group of nilpotency class two|Get more facts about aut-abelian group
Statement
A group of nilpotency class two need not be an aut-abelian group, i.e., its automorphism group need not be an abelian group.
Related facts
- Cyclic implies abelian automorphism group
- Abelian automorphism group implies class two
- Abelian automorphism group not implies abelian
Stronger facts
Proof
Example of the Klein four-group
Further information: Klein four-group, symmetric group:S3
Consider the Klein four-group: the direct product of two copies of the cyclic group of order two. It is an abelian group, hence it is a group of nilpotency class two.
On the other hand, its automorphism group is isomorphic to the symmetric group of degree three, which is not an abelian group.
Example of the dihedral group of order eight
Further information: dihedral group:D8
The dihedral group of order eight is a non-abelian group of nilpotency class two. Its automorphism group is also isomorphic to it, i.e., it is a dihedral group of order eight. Thus, we have a group of class two whose automorphism group is not abelian.