# Finite group implies cyclic iff every subgroup is characteristic

From Groupprops

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 (?):

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