Right-transitively homomorph-containing subgroup: Difference between revisions
| (5 intermediate revisions by the same user not shown) | |||
| Line 4: | Line 4: | ||
==Definition== | ==Definition== | ||
A [[subgroup]] <math> | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''right-transitively homomorph-containing subgroup''' if, whenever <math>K</math> is a [[homomorph-containing subgroup]] of <math>H</math>, <math>K</math> is also a homomorph-containing subgroup of <math>G</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 10: | Line 10: | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::subhomomorph-containing subgroup]] || contains every homomorphic image of every subgroup || [[subhomomorph-containing implies right-transitively homomorph-containing]] || [[right-transitively homomorph-containing not implies subhomomorph-containing]] || {{intermediate notions short|right-transitively homomorph-containing subgroup|subhomomorph-containing subgroup}} | |||
|- | |||
| [[Weaker than::order-containing subgroup]] || contains every subgroup whose order divides its order || (via subhomomorph-containing) || (via subhomomorph-containing) || {{intermediate notions short|right-transitivity homomorph-containing subgroup|order-containing subgroup}} | |||
|- | |||
| [[Weaker than::variety-containing subgroup]] || contains every subgroup of the whole group in the variety it generates || (via subhomomorph-containing) || (via subhomomorph-containing) || {{intermediate notions short|right-transitively homomorph-containing subgroup|variety-containing subgroup}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::homomorph-containing subgroup]] || contains every homomorphic image of itself || (obvious) || [[homomorph-containment is not transitive]] || {{intermediate notions short|homomorph-containing subgroup|right-transitively homomorph-containing subgroup}} | |||
|- | |||
| [[Stronger than::fully invariant subgroup]] || contains every image of itself under an endomorphism of the whole group || (via homomorph-containing) || (via homomorph-containing) || {{intermediate notions short|fully invariant subgroup|right-transitively homomorph-containing subgroup}} | |||
|- | |||
| [[Stronger than::characteristic subgroup]] || contains every image of itself under an automorphism of the whole group || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|right-transitively homomorph-containing subgroup}} | |||
|- | |||
| [[Stronger than::normal subgroup]] || contains every image of itself under an inner automorphism of the whole group || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|right-transitively homomorph-containing subgroup}} | |||
|} | |||
Latest revision as of 14:40, 9 March 2020
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed a right-transitively homomorph-containing subgroup if, whenever is a homomorph-containing subgroup of , is also a homomorph-containing subgroup of .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subhomomorph-containing subgroup | contains every homomorphic image of every subgroup | subhomomorph-containing implies right-transitively homomorph-containing | right-transitively homomorph-containing not implies subhomomorph-containing | |FULL LIST, MORE INFO |
| order-containing subgroup | contains every subgroup whose order divides its order | (via subhomomorph-containing) | (via subhomomorph-containing) | |FULL LIST, MORE INFO |
| variety-containing subgroup | contains every subgroup of the whole group in the variety it generates | (via subhomomorph-containing) | (via subhomomorph-containing) | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| homomorph-containing subgroup | contains every homomorphic image of itself | (obvious) | homomorph-containment is not transitive | |FULL LIST, MORE INFO |
| fully invariant subgroup | contains every image of itself under an endomorphism of the whole group | (via homomorph-containing) | (via homomorph-containing) | |FULL LIST, MORE INFO |
| characteristic subgroup | contains every image of itself under an automorphism of the whole group | (via fully invariant) | (via fully invariant) | |FULL LIST, MORE INFO |
| normal subgroup | contains every image of itself under an inner automorphism of the whole group | (via characteristic) | (via characteristic) | |FULL LIST, MORE INFO |