Wreath product

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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 AH. 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 Cayley's theorem, 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.

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[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 ≤ in.

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 ≤ in,

(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 sumE { 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)


If A itself is a permutation group, then the wreath product Awr(H,U) can also be given the structure of a permutation group in two standard ways. If A acts on X, then the two actions of the wreath product are:

  • The imprimitive wreath product action on X×U. If an element a of A takes x to y in its action on X, then when a is considered as an element in the uth component of AU then it acts as the identity on all (x,v) with vu and takes (x,u) to (y,u). The action of and element of H is similar, if h takes u to v, then in the wreath product action it takes (x,u) to (x,v). In other words, the set X×U is partitioned into blocks X×{u}, and the action of the uth copy of A is the same as the normal action of A on X, the action of the vth copy of A is trivial when vu, and the action of H is merely to permute the blocks.
  • The primitive wreath product action on XU. The elements of the set XU can be viewed as vectors with components indexed by u. An element a of A when considered to be in the uth component of AU acts on the vector by acting as the identity on all but the uth component where it acts normally as a in A on X. The elements of H simply permute the components of the vectors.

[edit] Examples

  • A nice example to work out is \mathbb{Z}\wr C_3.
  • C_2 \wr (S_n, n)

is isomorphic to the group of signed permutation matrices of degree n.

  • S_n \wr C_2 is isomorphic to the automorphism group of the complete bipartite graph on (n,n) vertices.
  • The Sylow p-subgroup of the symmetric group on p2 points is the wreath product C_p \wr C_p. The Sylow p-subgroup of the symmetric group on pn+1 points is the wreath product of Cp with the Sylow p-subgroup of the symmetric group on pn points, sometimes called the (n+1)-fold iterated wreath product of Cp. More generally, the Sylow p-subgroup of any symmetric group on finitely many points is a direct product of iterated wreath products of Cp.
  • Similarly, every maximal p-subgroup of the general linear group \operatorname{GL}(d,{\Bbb Z}) is conjugate in \operatorname{GL}(d,{\Bbb Q}) to a particular direct product of iterated wreath products of Cp.

[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.
  • The wreath product is associative in that there is a natural isomorphism between (G \wr H) \wr K and G \wr (H \wr K). Indeed, this isomorphism is an isomorphism of permutation representations when using the imprimitive action and is effected by the natural set isomorphism from (X × Y) × Z to X × (Y × Z). A natural isomorphism of permutation representations for a mixture of imprimitive and product actions is effected by the natural set isomorphism from (XY)Z to X(Y × Z).
  • The wreath product is not in general commutative, in that for most groups G \wr H is not isomorphic to H \wr G. Indeed, when G and H are finite these groups do not even usually have the same number of elements.
  • Every imprimitive permutation group G naturally defines a partition of the set into blocks. If A is the stabilizer of one of the blocks X, then the quotient group of G by the normal core of A can be identified as a transitive permutation group (H,U) on the set of blocks, U. A itself is a permutation group acting on the single block X. Using the blocks to identify the domain of G with X×U, there is a natural embedding of G into the imprimitive wreath product (A,X) \wr (H,U).

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