Group satisfying normalizer condition: Difference between revisions
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* [[Weaker than::Nilpotent group]]: It turns out that for a [[finitely generated group]], the two properties are equivalent. {{proofofstrictimplicationat|[[Nilpotent implies normalizer condition]]|[[Normalizer condition not implies nilpotent]]}} | * [[Weaker than::Nilpotent group]]: It turns out that for a [[finitely generated group]], the two properties are equivalent. {{proofofstrictimplicationat|[[Nilpotent implies normalizer condition]]|[[Normalizer condition not implies nilpotent]]}} | ||
* [[Weaker than::Group in which every subgroup is subnormal]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Stronger than::Gruenberg group]] | |||
* [[Stronger than::Locally nilpotent group]] | * [[Stronger than::Locally nilpotent group]] | ||
* [[Stronger than::Group having no proper abnormal subgroup]] | * [[Stronger than::Group having no proper abnormal subgroup]] | ||
Revision as of 17:54, 1 September 2008
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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
Symbol-free definition
A group is said to satisfy the normalizer condition, if it satisfies the following equivalent conditions:
- The normalizer of any proper subgroup properly contains it
- There is no proper self-normalizing subgroup
- Every subgroup is ascendant
Definition with symbols
A group is said to satisfy a normalizer condition if for any proper subgroup of , with the inclusion being strict (that is, is properly contained in its normalizer).
Groups satisfying the normalizer condition have been termed N-groups but the term N-group is also used for groups with a particular condition on normalizers of solvable subgroups.
Relation with other properties
Stronger properties
- Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent. For proof of the implication, refer Nilpotent implies normalizer condition and for proof of its strictness (i.e. the reverse implication being false) refer Normalizer condition not implies nilpotent.
- Group in which every subgroup is subnormal
Weaker properties
- Gruenberg group
- Locally nilpotent group
- Group having no proper abnormal subgroup
- Group in which every maximal subgroup is normal
Metaproperties
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