# Cyclic group of prime power order

## Definition

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

1. It is cyclic and its order is a power of a prime.
2. It is isomorphic to the group of integers modulo a power of a prime.
3. 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 $G$ is a cyclic group of prime power order, and $H$ is a subgroup of $G$, then $H$ is also cyclic of prime power order.
quotient-closed group property Yes If $G$ is a cyclic group of prime power order, and $H$ is a normal subgroup of $G$, then $G/H$ is also cyclic of prime power order.
finite direct product-closed group property No See next column It is possible to have $G_1, G_2$ both cyclic groups of prime power order, such that the external direct product $G_1 \times G_2$ 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 $G_1, G_2$ 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