Intermediately endomorphism kernel

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]

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Definition

Let $G$ be a group and $H$ be a subgroup of $G<math>.Wesaythat<math>H$ is intermediately (an) endomorphism kernel if for any intermediate subgroup $K$ (with $H\le K\le G$, $H$ is an endomorphism kernel in $K$.

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