Intermediately endomorphism kernel

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Definition

Let $G$ be a group and $H$ be a subgroup of $G$. We say that $H$ is intermediately (an) endomorphism kernel if for any intermediate subgroup $K$ (with $H\leq K\leq G$, $H$ is an endomorphism kernel in $K$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	can use some of the same examples as for endomorphism kernel is not transitive	We can have groups $H\leq K\leq G$ such that $H$ is intermediately an endomorphism kernel in $K$ and $K$ is intermediately an endomorphism kernel in $G$ but $H$ is not intermediately an endomorphism kernel in $G$.
intermediate subgroup condition	Yes	obvious reasons	If $H\leq K\leq G$ are groups such that $H$ is intermediately an endomorphism kernel in $G$, then $H$ is intermediately an endomorphism kernel in $K$.
quotient-transitive subgroup property	Yes		If $H\leq K\leq G$ are groups such that $H$ is intermediately an endomorphism kernel in $G$ and $K/H$ is intermediately an endomorphism kernel in $G/H$, then $K$ is intermediately an endomorphism kernel in $G$.
trim subgroup property	Yes		In any group, the whole group and the trivial subgroup are intermediately endomorphism kernels.

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
subgroup of finite abelian group		follows from its being an endomorphism kernel, which in turn follows from subgroup lattice and quotient lattice of finite abelian group are isomorphic		|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