Group satisfying normalizer condition: Difference between revisions
m (N-group moved to Normalizer condition: N-group has another meaning) |
|||
| Line 9: | Line 9: | ||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] | 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 subgroup|proper]] [[self-normalizing subgroup]] | |||
* Every subgroup is [[ascendant subgroup|ascendant]] | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
Revision as of 17:26, 7 March 2008
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 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 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 termed a N-group or 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).
Relation with other properties
Stronger properties
- Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent.
Weaker properties
Metaproperties
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]