Wreath product of groups of order p
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
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Let be a prime number. The wreath product of groups of order is any of the following equivalent things:
- It is the wreath product of the cyclic group of order with the cyclic group of order , where the latter is given the regular action on a set of size .
- It is the semidirect product of the elementary abelian group of order and a cyclic group of order acting on it by cyclic permutation of coordinates.
- It is the -Sylow subgroup of the symmetric group of order .
|Value of prime||Value (order of the wreath product)||Wreath product of groups of order||Symmetric group of order , in which it is a -Sylow subgroup|
|2||8||dihedral group:D8||symmetric group:S4 (see D8 in S4)|
|3||81||wreath product of Z3 and Z3||symmetric group:S9|
|5||15625||wreath product of Z5 and Z5||symmetric group:S25|
|Function||Value||Similar groups||Explanation for function value|
|order||As a wreath product, it has order .|
|prime-base logarithm of order|
|exponent||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]|
|prime-base logarithm of exponent||2|
|nilpotency class||The group is a maximal class group.|
|derived length||2||The base of the semidirect product is an abelian normal subgroup with abelian quotient group.|