Classification of groups of prime-fourth order
Let be a prime number. Then, the groups of order can be classified as follows, with slightly different classifications for the cases , , and . The case is different even in so far as the number of possible groups is concerned. The cases and have minor differences with each other.
|Group property||Number of isomorphism classes of groups case , so order||Number of isomorphism classes of groups case odd prime|
|nilpotency class exactly two, i.e., class two group that is non-abelian||6||6|
|nilpotency class exactly three||3||4|
The five abelian groups
The nature and classification of the five abelian groups of order is the same for both the and odd cases; the abelian groups are classified by the set of unordered integer partitions of the number 4.
|Partition of 4||Corresponding abelian group (in general)||Corresponding abelian group case||GAP ID (2nd part) case||GAP ID (2nd part) case odd|
|4||cyclic group of prime-fourth order||cyclic group:Z16||1||1|
|3 + 1||direct product of cyclic group of prime-cube order and cyclic group of prime order||direct product of Z8 and Z2||5||5|
|2 + 2||direct product of cyclic group of prime-square order and cyclic group of prime-square order||direct product of Z4 and Z4||2||2|
|2 + 1 + 1||direct product of cyclic group of prime-square order and elementary abelian group of prime-square order||direct product of Z4 and V4||10||11|
|1 + 1 + 1 + 1||elementary abelian group of prime-fourth order||elementary abelian group:E16||14||15|