Wreath product
From Wikipedia, the free encyclopedia
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 interesting examples of groups.
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). By a theorem of Cayley, every group H is a transitive permutation group when acting on itself; therefore, the former case 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.
Contents |
[edit] Definition
Our first example is the wreath product of a group A and a finite group H. By Cayley's theorem, we may regard H as a subgroup of the symmetric group Sn for some positive 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).
The unrestricted wreath product is a semidirect product of G by H, defined by taking 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 )=( f · (h * 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)). In particular, H and U need not be finite. 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 )=( f · (h * 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 )=( f · (h * g), hk)
[edit] Examples
A nice example to work out is Z wr C3 ...
C2 wr (Sn, n) is isomorphic to the group of signed permutation matrices of degree n.
Sn wr C2 is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.
[edit] 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