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.
Table of contents |
2 Examples 3 Properties |
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 of g of G = An, we let h permute the components of g; i.e. we define that for all 1 ≤ i ≤ n,
Definition
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:
As before, define the action of h in H on g in G by
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
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.