Commutator-closed subgroup property

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 $$H,K \le G$$ such that $$H$$ and $$K$$ both satisfy property $$p$$ in $$G$$, the commutator $$[H,K]$$ also satisfies property $$p$$ in $$G$$.

Stronger metaproperties

 * Weaker than::Endo-invariance property:

Related metaproperties

 * Intersection-closed subgroup property
 * Join-closed subgroup property