wreath product

Let A and B be groups, and let B act on the set Γ. Define the action of B on the direct productPlanetmathPlanetmathPlanetmathPlanetmath AΓ by


for any fAΓ and γΓ. The wreath product of A and B according to the action of B on Γ, denoted AΓB, is the semidirect product of groups AΓB.

Let us pause to unwind this definition. The elements of AΓB are ordered pairs (f,b), where fAΓ and bB. The group operationMathworldPlanetmath is given by


Note that by definition of the action of B on AΓ,


The action of B on Γ in the semidirect product permutes the elements of a tuple fAΓ, and the group operation defined on AΓ gives pointwise multiplication. To be explicit, suppose Γ is an n-tuple, and let (f,b),(f,b)AΓB. Let bi denote b-1(i). Then

(f,b)(f,b) = ((f(1),f(2),,f(n)),b)((f(1),f(2),,f(n)),b)
= ((f(1),f(2),,f(n))(f(b1),f(b2),,f(bn)),bb)(*)
= ((f(1)f(b1),f(2)f(b2),,f(n)f(bn)),bb).

Notice the permutationMathworldPlanetmath of the indices in (*).

A bit amount of thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.

Title wreath product
Canonical name WreathProduct
Date of creation 2014-12-30 11:23:20
Last modified on 2014-12-30 11:23:20
Owner mps (409)
Last modified by juanman (12619)
Numerical id 21
Author mps (12619)
Entry type Definition
Classification msc 20E22
Defines wreath product