# Characteristicity is commutator-closed

From Groupprops

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)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about commutator-closed subgroup property

## Contents

## Statement

### Statement with symbols

Suppose is a group and are characteristic subgroups of (possibly equal). Then, the commutator , defined as the subgroup of generated by commutators between elements of and elements of , is also a characteristic subgroup of .

## 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:
- Characteristicity is closed under all
*deterministic*operations. This is because these*deterministic*operations commute with automorphisms. For instance:

### Analogues in other algebraic structures

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