Group of prime power order

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter
View other prime-parametrized group properties | View other group properties

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

Definition

A group of prime power order is defined as a finite group whose order is a power of a prime.

Relation with other properties

Weaker properties

• Nilpotent group: For full proof, refer: Prime power order implies nilpotent
• Solvable group
• Group whose order has at most two prime factors

Distribution of orders

Groups of prime order

For every prime $p$, there is only one group of order $p$, viz the cyclic group of $p$ elements.

Groups of prime-squared order

Any group whose order is the square of a prime must be Abelian. For full proof, refer: Prime squared is Abelianness-forcing

Hence there are two possibilities for such a group: the cyclic group of order $p^2$ and the elementary Abelian group of order $p^2$.

Groups of prime-cubed order

For the order a cube of a prime, there are three Abelian possibilities (corresponding to the three possible unordered partitions of 3). There are two non-Abelian possibilities, the group of unipotent matrices of order 3 over the prime field and the semidirect product of the cyclic group of order $p^2$ by a cyclic group of order $p$.

For higher orders

The number of groups of order $16=2^4$ is 14. For an odd prime $p$, the number of groups of order $p^4$ is 15.

For higher powers of the prime, the number of groups of prime power order depends on the prime (in general, the larger the prime, the greater the number of groups).