# Wreath product of groups of order p

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## Definition

Let $p$ be a prime number. The wreath product of groups of order $p$ is any of the following equivalent things:

1. It is the wreath product of the cyclic group of order $p$ with the cyclic group of order $p$, where the latter is given the regular action on a set of size $p$.
2. It is the semidirect product of the elementary abelian group of order $p^p$ and a cyclic group of order $p$ acting on it by cyclic permutation of coordinates.
3. It is the $p$-Sylow subgroup of the symmetric group of order $p^2$.

## Particular cases

Value of prime $p$ Value $p^{p+1}$ (order of the wreath product) Wreath product of groups of order $p$ Symmetric group of order $p^2$, in which it is a $p$-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 $p^{p+1}$ As a wreath product, it has order $p^p \cdot p = p^{p+1}$.
prime-base logarithm of order $p + 1$
exponent $p^2$ PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
prime-base logarithm of exponent 2
nilpotency class $p$ 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 $p$ 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.