Derived subgroup is characteristic

Statement
Suppose $$G$$ is a group. Denote by $$G' = [G,G]$$ the derived subgroup of $$G$$, i.e., the subgroup generated by the commutators of pairs of elements of $$G$$. $$G'$$ is a characteristic subgroup of $$G$$.

Stronger facts

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

Proof idea
The proof idea is that for $$x,y \in G$$ and $$\sigma \in \operatorname{Aut}(G)$$:

$$\sigma([x,y]) = [\sigma(x),\sigma(y)]$$

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.