Characteristic direct factor of abelian group: Difference between revisions
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| [[Stronger than::direct factor of abelian group]] || || || || {{intermediate notions short|direct factor of abelian group|characteristic direct factor of abelian group}} | | [[Stronger than::direct factor of abelian group]] || || || || {{intermediate notions short|direct factor of abelian group|characteristic direct factor of abelian group}} | ||
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| [[Stronger than::fully invariant subgroup | | [[Stronger than::fully invariant subgroup]] || || || || {{intermediate notions short|fully invariant subgroup|characteristic direct factor of abelian group}} | ||
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| [[Stronger than::characteristic subgroup | | [[Stronger than::characteristic subgroup]] || || || || {{intermediate notions short|characteristic subgroup|characteristic direct factor of abelian group}} | ||
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| [[Stronger than::direct factor | | [[Stronger than::direct factor]] || || || || {{intermediate notions short|direct factor|characteristic direct factor of abelian group}} | ||
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Latest revision as of 20:15, 14 July 2013
This article describes a property that arises as the conjunction of a subgroup property: characteristic direct factor with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
This article describes a property that arises as the conjunction of a subgroup property: fully invariant direct factor with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
A subgroup of a group is termed a characteristic direct factor of if the following equivalent conditions are satisfied:
- is an abelian group and is a characteristic direct factor of (i.e., is both a characteristic subgroup of and a direct factor in ).
- is an abelian group and is a fully invariant direct factor of (i.e., is a fully invariant subgroup as well as a direct factor of ). See also equivalence of definitions of fully invariant direct factor for other equivalent formulations of this.
Equivalence of definitions
Further information: equivalence of definitions of characteristic direct factor of abelian group
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| characteristic direct factor of nilpotent group | |FULL LIST, MORE INFO | |||
| fully invariant direct factor | |FULL LIST, MORE INFO | |||
| characteristic direct factor | |FULL LIST, MORE INFO | |||
| fully invariant subgroup of abelian group | |FULL LIST, MORE INFO | |||
| characteristic subgroup of abelian group | |FULL LIST, MORE INFO | |||
| direct factor of abelian group | |FULL LIST, MORE INFO | |||
| fully invariant subgroup | |FULL LIST, MORE INFO | |||
| characteristic subgroup | |FULL LIST, MORE INFO | |||
| direct factor | |FULL LIST, MORE INFO |