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
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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

Symbol-free definition

A group of prime power order is defined in the following equivalent ways:

• It is a finite group whose order is a power of a prime.
• It is a finite group that is also a p-group for some prime $p$: the order of every element is a power of that same prime $p$

Equivalence of definitions

For full proof, refer: Equivalence of definitions of group of prime power order

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

Further information: enumeration of groups of prime power order

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).

Higman and Sims have studied in detail the function:

$f(n,p)$

which measures the number of isomorphism classes of groups of order $p^n$.