Nilpotent not implies nilpotent automorphism group: Difference between revisions
m (Vipul moved page Nilpotent not implies aut-nilpotent to Nilpotent not implies nilpotent automorphism group) |
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{{group property non-implication| | {{group property non-implication| | ||
stronger = nilpotent group| | stronger = nilpotent group| | ||
weaker = | weaker = group whose automorphism group is nilpotent}} | ||
==Statement== | ==Statement== | ||
The [[automorphism group]] of a [[nilpotent group]] need not be nilpotent. In other words, a nilpotent group is not necessarily an [[ | The [[automorphism group]] of a [[nilpotent group]] need not be nilpotent. In other words, a nilpotent group is not necessarily an [[group whose automorphism group is nilpotent]]. | ||
==Proof== | ==Proof== | ||
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{{further|[[particular example::Klein four-group]], [[particular example::symmetric group:S3]]}} | {{further|[[particular example::Klein four-group]], [[particular example::symmetric group:S3]]}} | ||
The automorphism group of the [[Klein four-group]], which is an abelian and hence nilpotent group, is the [[symmetric group:S3|symmetric group of degree three]], which is not a nilpotent group. | The [[automorphism group]] of the [[Klein four-group]], which is an abelian and hence nilpotent group, is the [[symmetric group:S3|symmetric group of degree three]], which is not a nilpotent group. | ||
Latest revision as of 19:03, 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., nilpotent group) need not satisfy the second group property (i.e., group whose automorphism group is nilpotent)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about nilpotent group|Get more facts about group whose automorphism group is nilpotent
Statement
The automorphism group of a nilpotent group need not be nilpotent. In other words, a nilpotent group is not necessarily an group whose automorphism group is nilpotent.
Proof
Example of the Klein four-group
Further information: Klein four-group, symmetric group:S3
The automorphism group of the Klein four-group, which is an abelian and hence nilpotent group, is the symmetric group of degree three, which is not a nilpotent group.