Characteristicity is commutator-closed

Statement with symbols
Suppose $$G$$ is a group and $$H,K$$ are characteristic subgroups of $$G$$ (possibly equal). Then, the commutator $$[H,K]$$, defined as the subgroup of $$G$$ generated by commutators between elements of $$H$$ and elements of $$K$$, is also a characteristic subgroup of $$G$$.

Generalizations and other particular cases

 * Endo-invariance implies commutator-closed: Any subgroup property arising as invariance under a collection of endomorphisms is a commutator-closed subgroup property. Other particular cases of this are:
 * Normality is commutator-closed
 * Strict characteristicity is commutator-closed
 * Full invariance is commutator-closed
 * Characteristicity is closed under all deterministic operations. This is because these deterministic operations commute with automorphisms. For instance:
 * Characteristicity is strongly intersection-closed
 * Characteristicity is strongly join-closed
 * Characteristicity is centralizer-closed

Analogues in other algebraic structures
Here are some analogues for Lie rings (equivalent statements apply for Lie algebras):


 * Characteristicity is Lie bracket-closed
 * Derivation-invariance is Lie bracket-closed