# Subgroup of cyclic group

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

Let be a group and a subgroup. We use the term **subgroup of cyclic group** to describe in if is a cyclic group.

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (Reverse implication failure) | Intermediate notions |
---|---|---|---|---|

verbal subgroup | cyclic implies every subgroup is verbal | (any non-cyclic group as a subgroup of itself) | |FULL LIST, MORE INFO | |

fully invariant subgroup | (via verbal) | (via verbal) | |FULL LIST, MORE INFO | |

characteristic subgroup | (via fully invariant) | (via fully invariant) | |FULL LIST, MORE INFO | |

normal subgroup | (via characteristic) | (via characteristic) | Characteristic subgroup, Subgroup of abelian group|FULL LIST, MORE INFO | |

subgroup of abelian group | the whole group is abelian | |FULL LIST, MORE INFO | ||

cyclic normal subgroup | cyclic and a normal subgroup | |FULL LIST, MORE INFO | ||

abelian normal subgroup | abelian and a normal subgroup | Subgroup of abelian group|FULL LIST, MORE INFO | ||

cyclic characteristic subgroup | cyclic and a characteristic subgroup | |FULL LIST, MORE INFO | ||

abelian characteristic subgroup | abelian and a characteristic subgroup | |FULL LIST, MORE INFO |