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