Subgroup whose normal closure is fully invariant

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Definition

A subgroup $H$ of a group $G$ is termed a subgroup whose normal closure is fully invariant if $H^{G}$, its normal closure in the whole group, is a fully invariant subgroup of $G$.

Examples

Extreme examples

• The trivial subgroup satisfies this property in every group.
• Every group satisfies this property in itself.

Important classes of examples

• All subgroups inside simple groups, as well as all contranormal subgroups, satisfy this property.
• Sylow subgroups, as well as subgroups arising as a join of Sylow subgroups (and in particular Hall subgroups) satisfy this property.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup	invariant under all endomorphisms			|FULL LIST, MORE INFO
endomorph-dominating subgroup	image under any endomorphism is contained in a conjugate subgroup to it			|FULL LIST, MORE INFO
Sylow subgroup	$p$-subgroup of finite group whose index is relatively prime to $p$	(via endomorph-dominating)	(via endomorph-dominating)	|FULL LIST, MORE INFO
join of Sylow subgroups	join of Sylow subgroups			|FULL LIST, MORE INFO
Hall subgroup	subgroup whose order and index are relatively prime	(via join of Sylow subgroups)	(via join of Sylow subgroups)	|FULL LIST, MORE INFO
subgroup whose normal closure is homomorph-containing	normal closure is a homomorph-containing 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 equals its own normal closure	|FULL LIST, MORE INFO
contranormal subgroup	normal closure is the whole group			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
closure-characteristic subgroup	normal closure is a characteristic subgroup	fully invariant implies characteristic	follows by combining characteristic not implies fully invariant and the fact that, since characteristic implies normal, a characteristic subgroup is its own normal closure	|FULL LIST, MORE INFO
subgroup whose characteristic closure is fully invariant	characteristic closure is a fully invariant subgroup	if the normal closure is fully invariant, it is also characteristic (because fully invariant implies characteristic), and since characteristic implies normal, it is the characteristic closure	proper nontrivial subgroup in a characteristically simple group, e.g., Z2 in V4	|FULL LIST, MORE INFO