Commutator-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Symbol-free definition

A subgroup property $p$ is termed commutator-closed if the commutator of any two subgroups, each of which has property $p$ in the whole group, also has property $p$ in the whole group.

Definition with symbols

A subgroup property $p$ is termed commutator-closed if, given any group $G$ and subgroups $H,K\leq G$ such that $H$ and $K$ both satisfy property $p$ in $G$, the commutator $[H,K]$ also satisfies property $p$ in $G$.

Relation with other metaproperties

Stronger metaproperties

• Endo-invariance property: For full proof, refer: Endo-invariance implies commutator-closed

Related metaproperties

• Intersection-closed subgroup property
• Join-closed subgroup property