Cyclic group of prime power order

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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