Group satisfying normalizer condition: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Weaker than::Nilpotent group]]: It turns out that for a [[finitely generated group]], the two properties are equivalent. | * [[Weaker than::Nilpotent group]]: It turns out that for a [[finitely generated group]], the two properties are equivalent. {{proofat|[[Nilpotent implies normalizer condition]]}} | ||
===Weaker properties=== | ===Weaker properties=== | ||
Revision as of 19:05, 20 May 2008
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 full proof, refer: Nilpotent implies normalizer condition
Weaker properties
Metaproperties
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