Trivial subgroup is characteristic

Statement
In any group, the trivial subgroup is a characteristic subgroup, i.e., any automorphism of the whole group maps the trivial subgroup to itself.

Proof
By definition, a homomorphism of groups must send the identity element to the identity element. Thus, any automorphism of a group must send its identity element to its identity element, and hence, must map the trivial subgroup to within itself.