T*-group: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is said to be a T*-group if | A [[group]] is said to be a T*-group if it satisfies the following equivalent conditions: | ||
* Every subgroup of it is a [[defining ingredient::T-group]] | |||
* If a subgroup is subnormal in some intermediate subgroup, it is also normal in that intermediate subgroup. | |||
* Every subgroup of it is an [[defining ingredient::intermediately subnormal-to-normal subgroup]]. | |||
===Definition with symbols=== | ===Definition with symbols=== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Abelian group]] | * [[Weaker than::Abelian group]] | ||
* [[Dedekind group]] | * [[Weaker than::Dedekind group]] | ||
* [[Weaker than::Group in which every subgroup is pronormal]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[T-group]] | * [[Stronger than::T-group]] | ||
* [[HN*-group]] | * [[Stronger than::HN*-group]] | ||
* [[HN-group]] | * [[Stronger than::HN-group]] | ||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 00:36, 22 February 2009
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 is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to T*-group, all facts related to T*-group) |Survey articles about this | Survey articles about definitions built on this
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View a list of other standard non-basic definitions
Definition
Symbol-free definition
A group is said to be a T*-group if it satisfies the following equivalent conditions:
- Every subgroup of it is a T-group
- If a subgroup is subnormal in some intermediate subgroup, it is also normal in that intermediate subgroup.
- Every subgroup of it is an intermediately subnormal-to-normal subgroup.
Definition with symbols
A group is said to be a T*-group if whenever , we have . In other words, any subgroup of is a T-group.
Relation with other properties
Stronger properties
Weaker properties
Metaproperties
Subgroups
This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties
Any subgroup of a T*-group is a T*-group. This follows from the definition.
Quotients
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties