Wreath product

In mathematics, the wreath product of group theory 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 constructing examples.
The standard or unrestricted wreath product of a group A by a group H is written as A_{ wr }H, or also A ≀ 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 componentwise multiplication; i.e., for any elements f, g in G, (f·g)_{i} = f_{i}g_{i} for 1 ≤ i ≤ n.
To specify the action "*" of an element h in H on an element g of G = A^{n}, 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 A^{n}, 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 multiplication is again defined as (f · g)_{u} = f_{u}g_{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 }C_{3} ...
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_{k}g' _{hk}}hh' for all k in H