# 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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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 $H \le K \le G$, $H$ 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 $H \le K \le G$ 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 $H \le K \le G$ 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 $H \le K \le G$ 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