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