Finite group implies cyclic iff every subgroup is characteristic

Statement
The following are equivalent for a fact about::finite group:


 * 1) It is a fact about::cyclic group, i.e., a finite cyclic group.
 * 2) Every subgroup of the group is a fact about::characteristic subgroup.
 * 3) Every subgroup of the group is a fact about::fully invariant subgroup.
 * 4) Every subgroup of the group is an fact about::isomorph-free subgroup.
 * 5) Every subgroup of the group is a fact about::verbal subgroup.
 * 6) Every subgroup of the group is a fact about::homomorph-containing subgroup.