Characteristicity is commutator-closed

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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

Analogues in other algebraic structures

Here are some analogues for Lie rings (equivalent statements apply for Lie algebras):