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Wreath ProductIn 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 Sn for some integer n. We start with the set G = A n, which is the cartesian product of n copies of A, each component xi 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 = figi for 1 ≤ i ≤ n. To specify the action "*" of an element h in H on an element 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 = gh -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 = AU of |U| copies of A. (If U is infinite, we take G to be the external direct sum ∑E { Au } of |U| copies of A, instead of the cartesian product). Pointwise multiplication is again defined as (f · g)u = fugu for all u in U. As before, define the action of h in H on g in G by - (h * g)u = gh -1(u)
and then define A wr (H, U) as the semidirect product of AU by H, with elements of the form (g, h) with g in AU, 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 = AH is - (h * g)k = gh -1k
and then define A wr H as the semidirect product of AH 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 ... C2 wr Sn 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 AH). 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 = AH 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 AH 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' = {gkg' hk}hh' for all k in H
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