Wreath product of groups of order p

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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 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
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.