Right-transitively homomorph-containing subgroup

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