# Cyclic group of prime power order

From Groupprops

## Contents

## Definition

A **cyclic group of prime power order** is a group satisfying the following equivalent conditions:

- It is cyclic and its order is a power of a prime.
- It is isomorphic to the group of integers modulo a power of a prime.
- It is either trivial or it is not generated by its proper subgroups.

### Equivalence of definitions

`Further information: Cyclic of prime power order iff not generated by proper subgroups`

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | If is a cyclic group of prime power order, and is a subgroup of , then is also cyclic of prime power order. | |

quotient-closed group property | Yes | If is a cyclic group of prime power order, and is a normal subgroup of , then is also cyclic of prime power order. | |

finite direct product-closed group property | No | See next column | It is possible to have both cyclic groups of prime power order, such that the external direct product is not cyclic of prime power order. In fact, any example where both groups are nontrivial works. |

lattice-determined group property | Yes | See next column | If have isomorphic lattices of subgroups, then either both are cyclic of prime power order, or neither is. Explicitly, the condition on the lattice of subgroups is that it is a path. |

## Relation with other properties

### Stronger properties

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

group of prime order | order equals a prime number | |FULL LIST, MORE INFO |

### Weaker properties

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

finite cyclic group | |FULL LIST, MORE INFO | |||

abelian group of prime power order | |FULL LIST, MORE INFO |