Finite group implies cyclic iff every subgroup is characteristic

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This article gives a proof/explanation of the equivalence of multiple definitions for the term finite cyclic group
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a Finite group (?):

  1. It is a Cyclic group (?), i.e., a finite cyclic group.
  2. Every subgroup of the group is a Characteristic subgroup (?).
  3. Every subgroup of the group is a Fully invariant subgroup (?).
  4. Every subgroup of the group is an Isomorph-free subgroup (?).
  5. Every subgroup of the group is a Verbal subgroup (?).
  6. Every subgroup of the group is a Homomorph-containing subgroup (?).