Intermediately endomorphism kernel: Difference between revisions
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==Definition== | ==Definition== | ||
Let <math>G</math> be a [[group]] and <math>H</math> be a [[subgroup]] of <math>G<math>. We say that <math>H</math> is '''intermediately''' (an) '''endomorphism kernel''' if for any intermediate subgroup <math>K</math> (with <math>H \le K \le G</math>, <math>H</math> is an [[defining ingredient::endomorphism kernel]] in <math>K</math>. | Let <math>G</math> be a [[group]] and <math>H</math> be a [[subgroup]] of <math>G</math>. We say that <math>H</math> is '''intermediately''' (an) '''endomorphism kernel''' if for any intermediate subgroup <math>K</math> (with <math>H \le K \le G</math>, <math>H</math> is an [[defining ingredient::endomorphism kernel]] in <math>K</math>. | ||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 19:27, 16 February 2013
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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
Definition
Let be a group and be a subgroup of . We say that is intermediately (an) endomorphism kernel if for any intermediate subgroup (with , is an endomorphism kernel in .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| complemented normal subgroup | normal and has a permutable complement, i.e., part of an internal semidirect product | follows from complemented normal satisfies intermediate subgroup condition and complemented normal implies endomorphism kernel | |FULL LIST, MORE INFO | |
| direct factor | (via complemented normal) | (via complemented normal) | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| endomorphism kernel | kernel of an endomorphism | (by definition) | follows from endomorphism kernel does not satisfy intermediate subgroup condition | |FULL LIST, MORE INFO |
| normal subgroup | (via endomorphism kernel) | (via endomorphism kernel) | |FULL LIST, MORE INFO |
Formalisms
In terms of the intermediately operator
This property is obtained by applying the intermediately operator to the property: endomorphism kernel
View other properties obtained by applying the intermediately operator