Closure-characteristic subgroup

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Definition

QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic

A subgroup of a group is termed closure-characteristic, or a subgroup whose normal closure is characteristic, if its normal closure in the whole group is a characteristic subgroup. In symbols, a subgroup $H$ of a group $G$ is termed closure-characteristic if the normal closure $H^G$ of $H$ in $G$ is characteristic in $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
automorph-dominating subgroup	all automorphic subgroups are contained in conjugate subgroups			|FULL LIST, MORE INFO
automorph-conjugate subgroup	all automorphic subgroups are conjugate	(via automorph-dominating)		|FULL LIST, MORE INFO
join of automorph-conjugate subgroups	join of automorph-conjugate subgroups			|FULL LIST, MORE INFO
Sylow subgroup	$p$-subgroup of finite group with index relatively prime to $p$			|FULL LIST, MORE INFO
join of Sylow subgroups	join of Sylow subgroups			|FULL LIST, MORE INFO
Hall subgroup	subgroup of finite group whose order and index are relatively prime			|FULL LIST, MORE INFO
contranormal subgroup	normal closure is whole group			|FULL LIST, MORE INFO

Conjunction with other properties

Any normal subgroup that is also closure-characteristic, is characteristic.

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
trim subgroup property	Yes	(obvious)	0	In any group $G$, both the trivial subgroup $\{e\}$ and the whole group $G$ are closure-characteristic.
strongly join-closed subgroup property	Yes	closure-characteristicity is strongly join-closed		Suppose $G$ is a group and $H_i,i\in I$ are all closure-characteristic subgroups of $G$. Then the join $⟨H_i{⟩}_{i\in I}$ is also closure-characteristic.