# Derived subgroup is characteristic

From Groupprops

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) always satisfies a particular subgroup property (i.e., characteristic subgroup)}

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

Suppose is a group. Denote by the derived subgroup of , i.e., the subgroup generated by the commutators of pairs of elements of . is a characteristic subgroup of .

## Related facts

### Stronger facts

- Derived subgroup is verbal combined with verbal implies fully invariant and fully invariant implies characteristic

## Proof

### Proof idea

The proof idea is that for and :

Thus, the set of elements expressible as commutators is invariant under any automorphism. Hence, the subgroup generated by this set is also invariant under any automorphism.

### Proof details

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]