Cyclic implies every subgroup is characteristic

Statement
In a cyclic group, every subgroup is a characteristic subgroup.

Stronger facts

 * Cyclic implies every subgroup is quotient-isomorph-free
 * Cyclic implies every subgroup is verbal
 * Finite cyclic implies every subgroup is isomorph-free
 * Locally cyclic and periodic implies every subgroup is characteristic

Converse

 * Finite and every subgroup is characteristic implies cyclic