Subgroup whose normal closure is homomorph-containing

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Definition

A subgroup $H$ of a group $G$ is termed a subgroup whose normal closure is homomorph-containing if $H^{G}$, its normal closure in the whole group $G$, is 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
homomorph-dominating subgroup	any subgroup of the whole group that is a homomorphic image of the subgroup is contained in one of its conjugates			|FULL LIST, MORE INFO
Sylow subgroup	subgroup of finite group with maximal prime power order	(via homomorph-dominating)		|FULL LIST, MORE INFO
join of Sylow subgroups	join of Sylow subgroups	follows from statement for Sylow subgroups, and homomorph-containment is strongly join-closed		|FULL LIST, MORE INFO
Hall subgroup	order and index are relatively prime	(via join of Sylow subgroups)		|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup whose normal closure is fully invariant	normal closure is a fully invariant subgroup	follows from homomorph-containing implies fully invariant	follows from fully invariant not implies homomorph-containing, and the fact that since fully invariant implies normal, a fully invariant subgroup is its own normal closure	|FULL LIST, MORE INFO
subgroup whose normal closure is characteristic	normal closure is a characteristic subgroup	(via subgroup whose normal closure is fully invariant, using fully invariant implies characteristic)	(via subgroup whose normal closure is fully invariant)	|FULL LIST, MORE INFO