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
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
Classification
Further information: Classification of groups of prime power order
Groups of prime order
For every prime , there is only one group of order , viz the cyclic group of 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 and the elementary Abelian group of order .
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 by a cyclic group of order .