Monotone convergence theorem

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.

Given two groups A and H there exist two variations of the wreath product: the unrestricted wreath product A Wr H (also written A≀H) and the restricted wreath product A wr H. Given a set Ω with an H-action there exists a generalisation of the wreath product which is denoted by A Wr_Ω H or A wr_Ω H respectively.

Definition

Let A and H be groups and Ω a set with H acting on it. Let K be the direct product

$K\equiv \prod _{\omega \in \mathrm{\Omega }}{A}_{\omega }$

of copies of A_ω := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences (a_ω) of elements of A indexed by Ω with component wise multiplication. Then the action of H on Ω extends in a natural way to an action of H on the group K by

$h(a_{\omega })\equiv (a_{h^{-1}\omega })$.

Then the unrestricted wreath product A Wr_Ω H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr_Ω H is called the base of the wreath product.

The restricted wreath product A wr_Ω H is constructed in the same way as the unrestricted wreath product except that one uses the direct sum

$K\equiv \underset{\omega \in \mathrm{\Omega }}{⨁}{A}_{\omega }$

as the base of the wreath product. In this case the elements of K are sequences (a_ω) of elements in A indexed by Ω of which all but finitely many a_ω are the identity element of A.

The group H acts in a natural way on itself by left multiplication. Thus we can choose Ω := H. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively. We say in this case that the wreath product is regular.

Notation and Conventions

The structure of the wreath product of A by H depends on the H-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention on the circumstances.

• In literature A≀_ΩH may stand for the unrestricted wreath product A Wr_Ω H or the restricted wreath product A wr_Ω H.

• Similarly, A≀H may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H.

• In literature the H-set Ω may be omitted from the notation even if Ω≠H.

• In the special case that H = S_n is the symmetric group of degree n it is common in the literature to assume that Ω={1,...,n} (with the natural action of S_n) and then omit Ω from the notation. That is, A≀S_n commonly denotes A≀_{1,...,n}S_n instead of the regular wreath product A≀_SnS_n. In the first case the base group is the product of n copies of A, in the latter it is the product of n! copies of A.

Properties

• Since the finite direct product is the same as the finite direct sum of groups it follows that the unrestricted A Wr_Ω H and the restricted wreath product A wr_Ω H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.

• A wr_Ω H is always a subgroup of A Wr_Ω H.

• Universal Embedding Theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.

• If A, H and Ω are finite, then

|A≀_ΩH| = |A|^|Ω||H|.

Canonical Actions of Wreath Products

If the group A acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which A Wr_Ω H (and therefore also A wr_Ω H) can act.

• The imprimitive wreath product action on Λ×Ω.

If ((a_ω),h)∈A Wr_Ω H and (λ,ω')∈Λ×Ω, then

$((a_{\omega }),h)\cdot (\lambda ,{\omega }^{\prime }):=(a_{h({\omega }^{\prime })}\lambda ,h{\omega }^{\prime })$.

• The primitive wreath product action on Λ^Ω.

An element in Λ^Ω is a sequence (λ_ω) indexed by the H-set Ω. Given an element ((a_ω), h) ∈ A Wr_Ω H its operation on (λ_ω)∈Λ^Ω is given by

$((a_{\omega }),h)\cdot ({\lambda }_{\omega }):=(a_{h^{-1}\omega }{\lambda }_{h^{-1}\omega })$.

Examples

• The Lamplighter group is the restricted wreath product ℤ₂≀ℤ.

• ℤ_m≀S_n (Generalized symmetric group).

The base of this wreath product is the n-fold direct product

ℤ_mⁿ = ℤ_m × ... × ℤ_m

of copies of ℤ_m where the action φ : S_n → Aut(ℤ_mⁿ) of the symmetric group S_n of degree n is given by

φ(σ)(α₁,..., α_n) := (α_σ(1),..., α_σ(n)).

• S₂≀S_n (Hyperoctahedral group).

The action of S_n on {1,...,n} is as above. Since the symmetric group S₂ of degree 2 is isomorphic to ℤ₂ the hyperoctahedral group is a special case of a generalized symmetric group.

• Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group S_pⁿ of degree pⁿ. Then P is isomorphic to the iterated regular wreath product W_n = ℤ_p ≀ ℤ_p≀...≀ℤ_p of n copies of ℤ_p. Here W₁ := ℤ_p and W_k := W_k-1≀ℤ_p for all k≥2.

• The Rubik's Cube group is a subgroup of small index in the product of wreath products, (ℤ₃≀S₈) × (ℤ₂≀S₁₂), the factors corresponding to the symmetries of the 8 corners and 12 edges.

References

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External links

• PlanetMath page
• Springer Online Reference Works
• Some Applications of the Wreath Product Construction