Wreath product

In group theory, the wreath product is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups; and also provide a way of generating new groups with specific desirable properties.

The standard or unrestricted wreath product of a group A by a group H is written as A _wr H. In addition, a more general version of the product can be defined for a group A and a transitive permutation group H acting on a set U, written as A _wr (H, U). Since by Cayley, every group H is a transitive permutation group when acting on itself, the former is a particular example of the latter.

An important distinction between the wreath product of groups A and H, and other products such as the direct sum, is that the actual product is a semidirect product of multiple copies of A by H, where H acts to permute the copies of A among themselves.

Definition

Our first example is the wreath product of a group A and a group H, where H is a subgroup of the symmetric group S_n for some integer n.

We start with the set G = A ⁿ, which is the cartesian product of n copies of A, each component x_i of an element x being indexed by [1,n]. We give this set a group structure by defining the group operation " · " as component-wise multiplication; i.e., for any elements f, g in G, (f·g)_i = f_ig_i for 1 ≤ i ≤ n.

To specify the action "*" of an element h in H on an element of g of G = Aⁿ, we let h permute the components of g; i.e. we define that for all 1 ≤ i ≤ n,

(h*g)_i = g_h ^-1(i)

In this way, it can be seen that each h induces an automorphism of G; i.e., h*(f · g) = (h*f) · (h*g). We can then define a semidirect product of G by H as follows: Define the unrestricted wreath product A _wr (H, n) as the set of all pairs { (g,h) | g in Aⁿ, h in H } with the following rule for the group operation:

( f, h )( g, k )=( (k * f) · g, hk)

More broadly, assume H to be any transitive permutation group on a set U (i.e., H is isomorphic to a subgroup of Sym(U)). The construction starts with a set G = A^U of |U| copies of A. (If U is infinite, we take G to be the external direct sum ∑_E { A_u } of |U| copies of A, instead of the cartesian product). Pointwise multipication is again defined as (f · g)_u = f_ug_u for all u in U.

As before, define the action of h in H on g in G by

(h * g)_u = g_h ^-1(u)

and then define A _wr (H, U) as the semidirect product of A^U by H, with elements of the form (g, h) with g in A^U, h in H and operation:

( f, h )( g, k )=( (k * f) · g, hk)

just as with the previous wreath product.

Finally, since every group acts on itself transitively, we can take U = H, and use the regular action of H on itself as the permutation group; then the action of h on g in G = A^H is

(h * g)_k = g_h ^-1k

and then define A _wr H as the semidirect product of A^H by H, with elements of the form (g, h) and again the operation:

( f, h )( g, k )=( (k * f) · g, hk)

Examples

A nice example to work out is Z _wr C₃ ...

C_{2 wr} S_n is isomorphic to the group of signed permutation matrices of degree n.

Properties

Every extension of A by H is isomorphic to a subgroup of A _wr H.

The elements of A _wr H are often written (g,h) or even gh (with g in A^H). First note that (e, h)(g, e) = (g, h), and (g, e)(e, h) = ((h*g), h). So (h ^-1*g, e)(e, h) = (g, h). Consider both G = A^H and H as actual subgroups of A _wr H by taking g for (g, e) and h for (e, h). Then for all g in A^H and h in H, we have that hg = (h ^-1*g)h.

The product (g,h)(g' ,h' ) is then easier to compute if we write (g,h)(g' ,h' ) as ghg'h'  and push g'  to the left using the commutative rule:

h {g' _k} = {g' _hk} h for all k in H

so that

ghg'h'  = {g_kg' _hk}hh'  for all k in H