# Right-transitively homomorph-containing subgroup

## Contents

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 $H$ of a group $G$ is termed a right-transitively homomorph-containing subgroup if, whenever $K$ is a homomorph-containing subgroup of $H$, $K$ is also a homomorph-containing subgroup of $G$.

## 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) Homomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup contains every image of itself under an automorphism of the whole group (via fully invariant) (via fully invariant) Homomorph-containing subgroup|FULL LIST, MORE INFO
normal subgroup contains every image of itself under an inner automorphism of the whole group (via characteristic) (via characteristic) Homomorph-containing subgroup|FULL LIST, MORE INFO