Difference between revisions of "Group in which every maximal subgroup is normal"

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(Relation with other properties)
(Stronger properties)
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Nilpotent group]]: {{proofat|[[Nilpotent implies every maximal subgroup is normal]]}}
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{| class="sortable" border="1"
* [[Weaker than::Hypercentral group]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Weaker than::Group satisfying normalizer condition]]: {{proofat|[[Normalizer condition implies every maximal subgroup is normal]]}}
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|-
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| [[Weaker than::nilpotent group]] || has a [[central series]] of finite length || [[nilpotent implies every maximal subgroup is normal]] || [[every maximal subgroup is normal not implies nilpotent]] || {{intermediate notions short|group in which every maximal subgroup is normal|nilpotent group}}
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|-
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| [[Weaker than::hypercentral group]] || [[hypercenter]] of the group (the point at which the transfinite upper central series stabilizes) is the whole group || [[hypercentral implies every maximal subgroup is normal]] || [[every maximal subgroup is normal not implies hypercentral]] || {{intermediate notions short|group in which every maximal subgroup is normal|hypercentral group}}
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|-
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| [[Weaker than::group satisfying normalizer condition]] || there is no proper [[self-normalizing subgroup]] || [[normalizer condition implies every maximal subgroup is normal]] || [[every maximal subgroup is normal not implies normalizer condition]] || {{intermediate notions short|group in which every maximal subgroup is normal|group satisfying normalizer condition}}
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|}

Revision as of 00:54, 26 December 2015

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

Definition

A group in which every maximal subgroup is normal is a group satisfying the following equivalent conditions:

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
nilpotent group has a central series of finite length nilpotent implies every maximal subgroup is normal every maximal subgroup is normal not implies nilpotent Group in which every subgroup is subnormal, Group satisfying normalizer condition|FULL LIST, MORE INFO
hypercentral group hypercenter of the group (the point at which the transfinite upper central series stabilizes) is the whole group hypercentral implies every maximal subgroup is normal every maximal subgroup is normal not implies hypercentral |FULL LIST, MORE INFO
group satisfying normalizer condition there is no proper self-normalizing subgroup normalizer condition implies every maximal subgroup is normal every maximal subgroup is normal not implies normalizer condition |FULL LIST, MORE INFO