Wreath product: Difference between revisions

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 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.

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 S_n 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 x_i 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 = f_ig_i 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 = 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).

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 = 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_ug_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 )=( 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 = 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 )=( f · (h * g), hk)

If A itself is a permutation group, then the wreath product A_wr(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 A^U then it acts as the identity on all (x,v) with v≠u 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 v≠u, and the action of H is merely to permute the blocks. Thus (f,h)(x,u)=(f_hu(x),h(u)).
• The primitive wreath product action on X^U. The elements of the set X^U can be viewed as vectors with components indexed by u. An element a of A when considered to be in the uth component of A^U 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.

• A nice example to work out is ${\displaystyle \mathbb {Z} \wr C_{3}}$.

• ${\displaystyle C_{2}\wr (S_{n},n)}$

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

• ${\displaystyle S_{n}\wr C_{2}}$ 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 ${\displaystyle C_{p}\wr C_{p}}$. The Sylow p-subgroup of the symmetric group on pⁿ⁺¹ points is the wreath product of C_p with the Sylow p-subgroup of the symmetric group on pⁿ points, sometimes called the (n+1)-fold iterated wreath product of C_p. More generally, the Sylow p-subgroup of any symmetric group on finitely many points is a direct product of iterated wreath products of C_p.

• Similarly, every maximal p-subgroup of the general linear group ${\displaystyle \operatorname {GL} (d,{\mathbb {Z}})}$ is conjugate in ${\displaystyle \operatorname {GL} (d,{\mathbb {Q}})}$ to a particular direct product of iterated wreath products of C_p.

• Every extension of A by H is isomorphic to a subgroup of ${\displaystyle A\wr H}$.

• The elements of ${\displaystyle A\wr H}$ 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 ${\displaystyle A\wr H}$ 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_kg' _hk}hh'  for all k in H.

• The wreath product is associative in that there is a natural isomorphism between ${\displaystyle (G\wr H)\wr K}$ and ${\displaystyle G\wr (H\wr K)}$. 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 (X^Y)^Z to X^{(Y × Z)}.

• The wreath product is not in general commutative, in that for most groups ${\displaystyle G\wr H}$ is not isomorphic to ${\displaystyle H\wr G}$. 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 ${\displaystyle (A,X)\wr (H,U)}$.

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