# Wreath product of groups of order p

From Groupprops

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.

View other such prime-parametrized groups

## Definition

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 .

## Particular cases

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 |

## Arithmetic functions

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. | |

Frattini length | 2 |

## GAP implementation

Assign to any numerical value of a prime number. Then, the group can be defined as ` WreathProduct(CyclicGroup(p),CyclicGroup(p))` using the GAP functions WreathProduct and CyclicGroup.