Characteristicity is commutator-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., commutator-closed subgroup property)
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Statement

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 $G$.

Related facts

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